[System]
You are a **scientific‐rigor auditor**. You will receive the parsed contents of a research paper. Your job is to identify **only** those errors or flaws that directly undermine the **scientific validity** of the paper’s methods, analyses, or conclusions.

Your sole focus is to identify flaws—such as errors in experimental design, data integrity, calculations, statistical inference, or reproducibility—that directly call into question the validity of a specific claim, paragraph, or the paper as a whole.

**Do not** report issues that are purely presentational, rhetorical, stylistic, or related to citation practices.

---

After you’ve done a **detailed walkthrough** of the paper, output exactly in this format—no extra keys or commentary:

```
<analysis>
{detailed walk‑through of how you checked each section/figure and why you flagged (or did not flag) any flaw}
</analysis>

<response>
{
  "has_error": <true|false>,
  "errors": [
    {
      "location": "Section 2.1",
      "description": "Claim that ‘all X are Y’ is contradicted by citation [5], which shows examples of non‑Y X."
    },
    {
      "location": "Figure 3",
      "description": "XAxis labeled ‘Time (s)’ but units appear to be milliseconds; scale bar mismatches caption."
    }
    // …more entries…
  ]
}
</response>
```

- **Do not** include any other keys or prose outside these two tagged blocks.
- **Do not** report stylistic or citation issues.
- Be as precise as possible about **where** (section ID or figure/table) and **what** the scientific flaw is.
- Each description for the errors must be rooted in a scientific rationale explaining why it is 'wrong' (not how it could be improved).

Begin your analysis now.

[User]
**Corrected OCR Text:**

---

Human pose estimation presents a critical challenge within the field of computer vision, bearing significant implications across a diverse array of applications, ranging from interactive gaming and virtual reality to clinical rehabilitation and security. Notwithstanding the substantial advancements catalysed by deep learning technologies, real-world environments' dynamic and often unpredictable nature continues to pose notable challenges. These encompass handling rapid movements, diverse postures, and intricate interactions within occluded environments.

The field of human pose estimation has witnessed remarkable progress, particularly with the advent of deep learning technologies. This section reviews the key advancements in this domain, highlights existing gaps in contemporary research, and justifies the present study's focus on addressing these challenges. By providing a comprehensive overview of the current landscape, this section lays the foundation for understanding the motivations and contributions of our work.

We appraise existing methodologies and their limitations when applied to the intricate scenarios that typify real-world applications. Subsequently, the paper elucidates the proposed enhancements to the MediaPipe framework, delineates our experimental setup, and discourses the findings from comprehensive tests. Ultimately, this research contributes to the broader domain of computer vision by furnishing a more robust solution that enhances the practical usability and scalability of human pose estimation technologies.

This study presents significant enhancements in human pose estimation using the MediaPipe framework. The research focuses on improving accuracy, computational efficiency, and real-time processing capabilities by comprehensively optimising the underlying algorithms. Novel modifications are introduced that substantially enhance pose estimation accuracy across challenging scenarios, such as dynamic movements and partial occlusions. The improved framework is benchmarked against traditional models, demonstrating considerable precision and computational speed gains. The advancements have wide-ranging applications in augmented reality, sports analytics, and healthcare, enabling more immersive experiences, refined performance analysis, and advanced patient monitoring. The study also explores the integration of these enhancements within mobile and embedded systems, addressing the need for computational efficiency and broader accessibility. The implications of this research set a new benchmark for real-time human pose estimation technologies and pave the way for future innovations in the field. The implementation code for the paper is available at https://github.com/avhixd/Human_pose_estimation

The rest of the paper is organised as follows. Section 2 provides the related work. Section 3 explores the methodology part. Sections 4 and 5 provide the experimental results and discussion along with the conclusion respectively.

MediaPipe, an open-source framework pioneered by Google, has surfaced as a potent tool offering robust, real-time, multi-person pose estimation capabilities. Nevertheless, its standard deployment necessitates further refinement to satisfy the exacting demands of real-time processing and high accuracy within complex operational environments.

Human pose estimation technologies have dramatically advanced due to deep learning. Early methods relied heavily on hand-crafted features and classical machine learning

The enhancements addressed existing lacunae and established a new benchmark in pose estimation capabilities, proffering insights into integrating such technologies in both extant and emerging markets.

This study is devoted to augmenting the MediaPipe framework by integrating sophisticated algorithmic enhancements and optimisations. Our objectives are twofold:

---

**Keywords:**
Human pose estimation
MediaPipe
Real-time processing
Computational efficiency and Dynamic motion capture

techniques, often limited to specific poses and environments. With the advent of deep learning, particularly convolutional neural networks (CNNs), more robust models like OpenPose Cao et al. (2019) and PoseNet Papandreou et al. (2018) have been developed. These models utilize vast datasets to learn rich representations of human anatomy, significantly improving accuracy in diverse settings Chen et al. (2020).

MediaPipe, introduced by Google, represents a pivotal shift towards lightweight, real-time frameworks designed for multi-person pose estimation on mobile devices. Its architecture allows for efficient processing, which is crucial for applications requiring instant feedback Zhang et al. (2020).

Despite the progress, several challenges persist in the domain of pose estimation. Current models perform well under controlled conditions but often falter in complex real-world scenarios characterized by rapid movements, varying lighting conditions, and occlusions Li et al. (2021). Moreover, while accuracy has been a primary focus, the computational efficiency necessary for deployment on low-power devices frequently remains unaddressed.

Furthermore, existing frameworks often overlook the integration of real-time feedback mechanisms, which are essential for interactive applications such as augmented reality and live sports analytics Wang et al. (2021).

This research addresses these gaps by advancing the MediaPipe framework and enhancing its robustness and computational efficiency. The proposed modifications aim to improve pose estimation accuracy in dynamically challenging environments while reducing the computational load to facilitate broader deployment, especially on mobile platforms.

Moreover, by enhancing the framework’s ability to handle real-time data processing, this study extends its applicability to fields that depend on immediate pose estimation, such as telemedicine and personal fitness apps, where quick and accurate feedback is critical Bazarevsky and Grishchenko (2020).

Summary—This section has reviewed key advancements and identified critical gaps in the field of human pose estimation, setting the stage for the subsequent presentation of this study’s contributions, which focus on overcoming these limitations to push the boundaries of what is currently achievable.

This section outlines the research design, system architecture, data collection, and analysis methods employed in this study. By detailing the experimental procedures and analytical techniques, we provide a comprehensive framework for understanding the enhancements made to the MediaPipe framework for human pose estimation and evaluating their effectiveness.

The study employs an experimental research design to rigorously assess the enhancements made to the MediaPipe framework for human pose estimation. This design aims to evaluate the improved accuracy and efficiency of pose estimation across diverse and dynamically changing scenarios. Experiments are designed to mimic real-world applications, providing a robust test environment for the optimized MediaPipe framework.

Figure 1 illustrates the architecture of the video capture and processing system used in this study. The system is divided into several key components:

Our pose estimation pipeline calculates the angle between three key points: shoulder, elbow, and wrist. The angle is computed using the arctangent function as follows:

angle = arctan2(c_y - b_y, c_x - b_x) - arctan2(a_y - b_y, a_x - b_x)

where (a_x, a_y), (b_x, b_y), and (c_x, c_y) represent the coordinates of the shoulder, elbow, and wrist landmarks, respectively. The resulting angle is converted from radians to degrees for further analysis.

This section presents the findings from the experimental evaluation of the optimized MediaPipe framework. The results are detailed through quantitative analysis, visual aids, curl counter logic, and a discussion of the findings, emphasizing the enhancements achieved in pose estimation accuracy and computational efficiency.

The MediaPipe pose estimation model returns a set of landmark coordinates representing key points on the human body. These landmarks are denoted as (x, y) coordinates, where x and y are normalized values between 0 and 1, representing the relative position within the image frame. The landmarks can be mapped to pixel coordinates by multiplying them with the image width and height:

pixelₓ = landmarkₓ × image width
pixelᵧ = landmarkᵧ × image height

These landmark coordinates are the basis for further analysis and calculations in the pose estimation pipeline.

Data collection leverages high-resolution cameras coupled with the MediaPipe framework, which is enhanced for higher precision and reduced latency in pose estimation. These tools are crucial for capturing accurate and dynamic movements of human subjects in diverse settings.

The data collection involves the systematic recording of human activities such as walking, running, and other dynamic movements in both controlled environments and natural settings with varying lighting conditions. The aim is to cover a comprehensive range of human poses and movement scenarios to ensure the system's robustness and applicability in real-world applications.

The accuracy and efficiency of the pose estimation are evaluated using advanced statistical methods. The primary performance indicators are:

MSE = (1/n) ∑_{i=1}^n (Y_i - Ŷ_i)^2     (1)

IoU = (Area of Overlap) / (Area of Union)      (2)

Additionally, computational efficiency metrics such as frame processing time and throughput are analyzed.

Qualitative assessments include visual inspections of pose estimation outcomes to evaluate system performance under real-world conditions. Feedback from test participants and system operators is integrated to assess user experience and responsiveness.

Table 1 summarizes the experimental data, indicating the system’s performance across various conditions:

Experimental results showing the performance of the pose estimation system under different environmental conditions.

These enhancements and experimental setups provide a thorough foundation for understanding and evaluating the practical applications and technological advancements achieved in the field of real-time human pose estimation.

The optimized MediaPipe model demonstrated significant enhancements in accuracy and computational efficiency. Key statistics include:

These improvements highlight the model’s capability to deliver high-performance pose estimation in real-time applications.

Figure 2 illustrates the reduction in average processing time per frame between the baseline MediaPipe model and the optimized version. This comparison emphasizes the effectiveness of the computational optimizations implemented.

Figure 3 demonstrates the application of the optimized MediaPipe framework in a real-time setting. This image showcases the system’s ability to accurately detect and label multiple key points on a human subject dynamically engaging in an activity. It exemplifies the system’s responsiveness and precision in a live scenario.

The enhancements in both accuracy and computational speed are attributed to the refined algorithms and the incorporation of advanced neural network architectures within the MediaPipe framework. These improvements facilitate rapid and precise pose estimations, which are vital for applications requiring real-time data processing, such as augmented reality and interactive gaming.

This section evaluates the broader implications, acknowledges the limitations of the current study, and outlines potential directions for future research, emphasizing the necessity and value of continual development in human pose estimation technologies.

Despite the promising results, this study is not without limitations. The experimental scenarios, though diverse, do not cover all possible real-world conditions, potentially limiting the generalizability of the findings. Additionally, the selection of datasets may introduce a bias, as certain types of movements or activities could be overrepresented. These factors must be carefully considered when interpreting the results and planning subsequent studies.

Looking ahead, it is crucial to extend the capabilities of the MediaPipe framework by integrating it with additional machine learning methodologies, such as reinforcement learning. This could potentially enable the system to adapt more dynamically to complex and unforeseen environments. Further research should also aim to reduce the computational resources required by the framework, thereby making it accessible on a wider range of devices, including those with lower processing power. Investigating these areas will help in overcoming the current limitations and enhance the framework's applicability and effectiveness in real-world applications.

The performance of the modified MediaPipe framework, particularly under challenging conditions like low light and partial occlusions, exhibits superior capabilities when compared to previous models documented in the literature. This suggests that the enhancements developed are effective in overcoming common limitations faced by existing pose estimation technologies, confirming the relevance and impact of this research in advancing the field of computer vision.

The improvements made to the MediaPipe framework have significant practical implications, enhancing real-time applications in fields such as augmented reality, healthcare, and sports. By achieving higher accuracy and reduced processing time, the system provides more immersive experiences in augmented reality and allows for more effective monitoring and analysis in healthcare and sports settings. This research thus contributes to the field of computer vision by addressing key challenges in pose estimation, particularly in environments that require robust performance under dynamic conditions.

The curl counter is incremented when transitioning from the "down" stage to the "up" stage. This logic enables accurate tracking and counting of curl repetitions.

In our curl counter logic, we determine the stage of the curl exercise based on the angle between the shoulder, elbow, and wrist. The stage is determined as follows:

As with any technological advancement, it is essential to consider the ethical implications of enhanced human

\[
\text{stage} =
\begin{cases}
\text{down}, & \text{if angle} > 160^\circ \\
\text{up}, & \text{if angle} < 30^\circ \text{ and previous\_stage} = \text{down}
\end{cases}
\]

pose estimation systems. While these technologies have the potential to revolutionize various domains, they also raise concerns about privacy, data security, and potential misuse. Future research should actively engage with these ethical questions, developing guidelines and safeguards to ensure that the benefits of these technologies are realized while minimizing risks and unintended consequences.

The advancements in human pose estimation presented in this study have far-reaching societal implications. In healthcare, improved pose tracking can lead to more effective rehabilitation techniques and early detection of movement disorders. In sports, it can provide athletes and coaches with detailed performance analysis and help prevent injuries. In the realm of entertainment, enhanced pose estimation can create more immersive and interactive experiences. However, it is crucial to consider the potential impact on employment, as automated systems may replace certain jobs that currently rely on human observation and analysis.

computational efficiency, and robustness of the system under various environmental conditions highlight its potential for real-world applications. However, the limitations and ethical considerations discussed in this section underscore the need for ongoing research and responsible development. As we continue to push the boundaries of what is possible with these technologies, we must remain committed to maximizing their benefits while navigating the challenges they present. By doing so, we can unlock their transformative potential and shape a future where human pose estimation technologies are not only technically sophisticated but also ethically sound and socially beneficial.

Our research achieved a notable improvement in the accuracy of human pose estimation, evidenced by a 20% increase in the Intersection over Union (IoU) metric compared to existing solutions. Furthermore, the enhancements led to a reduction in processing time by approximately 30%, affirming the effectiveness of the optimizations integrated into the MediaPipe framework. These advancements facilitate more responsive and precise applications in areas demanding real-time data processing, such as augmented reality and live sports analytics.

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

The data that has been used in this study is available from the corresponding author upon reasonable request.

The primary objective of this study was to refine and evaluate the MediaPipe framework to ensure it meets the demands of robust, real-time human pose estimation. Through rigorous testing and optimization, this work has substantially advanced the state of the art, setting a new benchmark for future research in the field.

Sandeep Singh Sengar: Conceptualization, Methodology, Supervision, Writing - review & editing. Abhishek Kumar: Methodology, Software, Writing - original draft. Owen Singh: Writing - review & editing.

Alam, E., Sufian, A., Dutta, P., Leo, M., 2022. Vision-based human fall detection systems using deep learning: A review. Comput. Biol. Med. 146, 105626.

Bazarevsky, V., Grishchenko, I., 2020. On-device, real-time body pose tracking with mediapipe blazepose. Google Research Blog URL: https://ai.googleblog.com/2020/08/on-device-real-time-body-pose-tracking.html.

Cao, Z., Simon, T., Wei, S.E., Sheikh, Y., 2019. Openpose: Realtime multi-person 2d pose estimation using part affinity fields. IEEE Transactions on Pattern Analysis and Machine Intelligence 32, 172–186.

Chen, Y., Tian, Y., He, M., 2020. Monocular human pose estimation: A survey of deep learning-based methods. Comput. Vis. Image Underst. 192, 102897.

Developer, B., 2020. Computational efficiency in pose estimation technologies: A neglected priority. IEEE Embedded Systems Letters 11, 49–52.

Innovator, C., 2021. Enhancing the robustness of pose estimation models in sport analytics applications. International Journal of Sports Technology 7, 34–45.

Kim, J.W., Choi, J.Y., Ha, E.J., Choi, J.H., 2023. Human pose estimation using mediapipe pose and optimization method based on a humanoid model. Applied Sciences 13, 2700.

Li, J., Xu, C., Chen, Z., Bian, S., Yang, L., Lu, C., 2021. Hybrik: A hybrid analytical-neural inverse kinematics solution for 3d human pose and shape estimation. Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, 3383–3393.

Papandreou, G., Zhu, T., Kanazawa, N., Toshev, A., 2018. Towards accurate multi-person pose estimation in the wild, in: Proceedings of the CVPR.

Pavllo, D., Feichtenhofer, C., Grangier, D., Auli, M., 2019. 3d human pose estimation in video with temporal convolutions and semi-supervised

Reflecting on the study, it is clear that while substantial progress has been made, the potential for further enhancements remains vast. Future work should focus on extending the framework's capabilities to handle more complex scenarios, including highly dynamic environments and varying lighting conditions. Additionally, integrating machine learning techniques such as deep reinforcement learning could offer adaptive improvements, making the system even more robust against diverse challenges encountered in real-world applications.

This study demonstrates the significant advancements achieved in human pose estimation through the optimization of the MediaPipe framework. The enhanced accuracy,

training. Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, 7753–7762.

Researcher, A., 2021. Integration of real-time feedback systems with pose estimation models: Challenges and prospects. Journal of Interactive Systems 12, 54–65.

Wang, J., Tan, S., Zhen, X., Xu, S., Zheng, F., He, Z., Shao, L., 2021. Deep 3d human pose estimation: A review. Elsevier, p. 103225.

Yurtsever, M., Eken, S., 2022. Babypose: Real-time decoding of baby's non-verbal communication using 2d video-based pose estimation. IEEE Sensors 22, 13776–13784.

Zhang, Y., et al., 2020. Optimizing mediapipe for real-time pose estimation on mobile devices. Journal of Mobile Computing 15, 288–299.


[System]
You are a **scientific‐rigor auditor**. You will receive the parsed contents of a research paper. Your job is to identify **only** those errors or flaws that directly undermine the **scientific validity** of the paper’s methods, analyses, or conclusions.

Your sole focus is to identify flaws—such as errors in experimental design, data integrity, calculations, statistical inference, or reproducibility—that directly call into question the validity of a specific claim, paragraph, or the paper as a whole.

**Do not** report issues that are purely presentational, rhetorical, stylistic, or related to citation practices.

---

After you’ve done a **detailed walkthrough** of the paper, output exactly in this format—no extra keys or commentary:

```
<analysis>
{detailed walk‑through of how you checked each section/figure and why you flagged (or did not flag) any flaw}
</analysis>

<response>
{
  "has_error": <true|false>,
  "errors": [
    {
      "location": "Section 2.1",
      "description": "Claim that ‘all X are Y’ is contradicted by citation [5], which shows examples of non‑Y X."
    },
    {
      "location": "Figure 3",
      "description": "XAxis labeled ‘Time (s)’ but units appear to be milliseconds; scale bar mismatches caption."
    }
    // …more entries…
  ]
}
</response>
```

- **Do not** include any other keys or prose outside these two tagged blocks.
- **Do not** report stylistic or citation issues.
- Be as precise as possible about **where** (section ID or figure/table) and **what** the scientific flaw is.
- Each description for the errors must be rooted in a scientific rationale explaining why it is 'wrong' (not how it could be improved).

Begin your analysis now.

[User]
Abstract. The conjecture of the existence and the uniqueness of the strictly convex quadrilateral central configuration for the Newtonian 4-body problem is one of the most-talked open problems in the study of the classical n-body problems in celestial mechanics. MacMillan and Bartky first gave its general existence in the 1930s and a particular case for its uniqueness. Still, the general case has yet to be solved perfectly since it was considered by Sim’o and Yoccoz in the 1980s and was first mentioned by Albouy and Fu in 2008 in the formal publication. Using coordinates of mutual distances and Morse’s critical point theory, we give the (at most) uniqueness of the planar strictly convex 4-body central configuration when the angle of one pair of the opposite sides is given.

The classical n-body problem mainly concentrates on the study of n particles P₁, ···, Pₙ with masses mᵢ > 0 and positions xᵢ = (xᵢ,₁,···,xᵢ,₋d)ᵗ ∈ ℝᵈ (i = 1,···,n; d = 2, 3) interacting with each other and satisfying the following differential equations

(1) mᵢ𝑥̈ᵢ,ⱼ = ∂U/∂xᵢ,ⱼ, i = 1,···,n, j = 1,···,d,

where

U = ∑_{i<j} (mᵢmⱼ)/|xⱼ − xᵢ| = ∑_{i<j} (mᵢmⱼ)/r_{i,j}

is the Newtonian potential and r_{i,j} = |xᵢ − xⱼ| is the mutual distance between Pᵢ and Pⱼ. We denote by x = (x₁,···,xₙ) ∈ ℝ^{dn} \ Δ a collision free configuration, where Δ = {x ∈ ℝ^{dn} | xᵢ = xⱼ, ∀ i ≠ j} is the singular set, or the collision set, and r = (r₁₂, r₁₃,···,r_{n–1,n}) ∈ (ℝ⁺)^{C²ₙ} its corresponding mutual distance vector.

YANGSHANSHAN LIU AND SHIQING ZHANG

Central configurations are particular arrangements of the $n$-body system satisfying a set of nonlinear algebraic equations

$$(2) \qquad -\lambda m_j(x_j - c) = \frac{\partial U(x)}{\partial x_j} = \sum_{\substack{i=1 \\ i \neq j}}^n \frac{m_i m_j (x_i - x_j)}{r_{ij}^3},$$

where $c = \frac{1}{m} \sum_{i=1}^n m_i x_i$ is the center of mass with $m = \sum_{i=1}^n m_i$ the total mass and $\lambda$ the configuration constant. Furthermore, if we denote by

$$(3) \qquad I(x) = \frac{1}{2} \sum_{i=1}^n m_i (x_i - c)^2 = \frac{1}{2m} \sum_{i<j} m_i m_j r_{ij}^2$$

the moment of inertia of the system, a more compact form of (2) is as follows

$$(4) \qquad \nabla U(x) + \lambda \nabla I(x) = 0,$$

in which $\nabla$ denotes the gradient operator with respect to $x$. Hence, $\lambda$ can be seen as the Lagrangian multiplier of the potential $U(x)$ with the constraint $I(x) = I_0$. For a more intuitive illustration, we can simultaneously release the $n$ particles with zero velocity from this special position, and all bodies will collide at the center of mass in a limited time. This can be seen as a special homothetic case of the homographic solutions of (1), i.e.,

$$x(t) - c(t) = r(t) Q(t) (x_0 - c_0),$$

where $x_0$ is a central configuration with $c_0$ its center of mass, $r(t) > 0$ is a real scaling factor, namely, a solution of Kepler's problem (i.e., 2-body problem), and $Q(t) \in SO(d)$ is a rotation. Furthermore, if we take $r(t) = 1$, only considering the rotation, we get the well-known relative equilibrium solutions. These self-similar solutions may be the only analytic solutions to the $n$-body problems up to the recent research. The first two kinds of homographic solutions were constructed by Euler in 1767 and Lagrange in 1772 with initial particular positions for the 3-body problem, respectively, which came to be known as the Eulerian collinear and Lagrangian equilateral triangle central configurations.

One can easily see that the equations (2) and (4) are invariant under translation, dilation, and rotation, which allows us to consider the equivalent classes of central configurations naturally. In addition, classical Morse's theory requires the non-degeneracy of central configurations as critical points at least, which are not isolated under the continuous action of the rotation group; hence, they are degenerate. Two central configurations are equivalent if they can be transformed from one to another by combining the above three transformations. There are several open problems concerning central configurations, one of which is the finiteness conjecture, namely, the Chazy-Wintner-Smale conjecture [6,33,35]: Is the number of equivalent classes of central configurations (or relative equilibria for $d = 2$) finite for any given $n$ positive masses? For $n = 3$, Wintner [35] showed that there are

totally three Eulerian collinear central configurations and two Lagrangian equilateral triangle central configurations. Hampton and Moeckel [15] gave a positive answer for the case \( n = 4 \), and it is still open for \( n \geq 5 \) with several generic results on \( n = 5 \), see [5, 14].

If we consider the particular shapes of the configurations, which are related to the classification of central configurations, some excellent results also come out. For example, Moulton [24] studied the \( n \) collinear case in 1910, concluding that there are just \( n! / 2 \) collinear central configurations for any given \( n \) positive masses along a straight line. One can also refer to [23, 32] for different proofs of this collinear case. In other words, if the order of the \( n \) masses along a line is fixed, the central configuration is unique. Similarly, the uniqueness of some non-collinear central configurations can also be obtained, provided the corresponding order of the masses is fixed. Related to this idea, one widely focused issue is the existence and uniqueness conjecture of the planar strictly convex 4-body central configuration. The existence property dates back to the work of MacMillan and Bartky [21] in 1932, and Xia [36] provided a concise proof in 2004 using perturbation techniques. One can also refer to Moeckel [23] for more details. For the uniqueness part, one can refer to the clear narrative by Santoprete in his recent papers [28, 29] that it is reasonable to be named after Simó and Yoccoz since, according to Alain Albouy, one can trace this uniqueness problem back to several conversations between Simó and Yoccoz, as well as their related works [31, 38].

Conjecture 1 (Simó-Yoccoz). Given four positive masses \( m_1, m_2, m_3, m_4 \), is there only one convex central configuration for each cyclic order?

Albouy and Fu first mentioned this conjecture in a formally published paper [1], and then Albouy, Cabral, and Santos collected it in [3] as Problem 10 on the classical n-body problems. There have been several excellent results around this issue, with different kinds of constraints on masses [2, 4, 7, 11, 18, 20, 26], shapes [19, 27–29, 37], potentials, and their combinations [8, 12, 13], and so on, using different analytical or topological approaches. Recently, Sun, Xie, and You [34] gave a computer-assisted proof in some generic cases using interval analysis based on the Krawczyk operator and the implicit function theorem, except for the ones with masses very close to 0.

In the current paper, we use a rigorous topological method inspired by the idea of Santoprete in [28, 29] to get the uniqueness result of the strictly convex planar 4-body central configuration with a given angle of one pair of opposite sides. We summarize the main steps as follows:

(i) Use mutual distances as variables instead of positions to eliminate the rotation symmetry naturally;
(ii) Replace the Cayley-Menger determinants constraint with a simple new one such that the gradients of both are paralleled;
(iii) Determine the type, i.e., the Morse index of the critical point of the reduced Lagrangian function;

For (ii), replacing the Cayley-Menger determinant constraints with more simple ones can significantly simplify the calculations in (iii), (iv) and (v), and was first noticed and raised by Cors and Roberts [9] in the study of co-circular central configurations of the 4-body problem in which they used Ptolemy’s theorem instead, supported by an important property, i.e., the gradients of both constraints with respect to \( r \) are parallel. Santoprete in [27] succeeded in replacing with a trapezoidal geometrical constraint. In this current paper, we use a necessary condition for a planar convex quadrilateral given by Josefsson in [16] (Theorem 10), which plays a crucial role in (ii), and we will see in the following.

We need to point out that according to the famous Perpendicular Bisector Theorem in [22], a convex planar 4-body central configuration must be strictly convex, i.e., any one of the four bodies is located outside the convex hull of the left three. In other words, there is no 3-body collinearity in a planar 4-body case and no \((n-1)\)-body collinearity in a planar n-body central configuration. Before going into the main steps, some notations are necessary.

Finally, we have

**Theorem 1.** *Given four positive masses \( m_1, m_2, m_3, m_4 \), there is at most only one convex central configuration for each cyclic order with a given angle of one pair of opposite sides.*

Let
\[
x = (x_1, \cdots, x_4), \quad x_i = (x_{i1}, x_{i2}) \in \mathbb{R}^2, \quad i = 1, \cdots, 4
\]
be the position vector and
\[
r = (r_{12}, r_{13}, \cdots, r_{34}) \in (\mathbb{R}^+)^6
\]
its distance vector, where
\[
P_i P_j = r_{ij} = |x_j - x_i| = \sqrt{(x_{j1} - x_{i1})^2 + (x_{j2} - x_{i2})^2}, \quad 1 \leq i < j \leq 4.
\]

The Cayley-Menger determinant is
\[
\mathcal{F}(1,2,\cdots,n) = F_{12\cdots n}(r) = F_n(1,2,\cdots, n)
\]
\[
= \begin{vmatrix}
0 & 1 & 1 & \cdots & 1 \\
1 & 0 & r_{12}^2 & \cdots & r_{1n}^2 \\
1 & r_{12}^2 & 0 & \cdots & r_{2n}^2 \\
\vdots & \vdots & \vdots & \ddots & \vdots \\
1 & r_{1n}^2 & r_{2n}^2 & \cdots & 0 \\
\end{vmatrix}
= (-1)^n \cdot 2^{n-1} \cdot ((n-1)!)^2 \cdot \mathcal{V}_n^2,
\]

where $\mathcal{V}_n$ is the volume of the simplex formed by $n$ points. Dziobek used it in [10] in 1900 as a planar constraint $F_4(r) = 0$ to characterize the equations of central configurations of the 4-body problem and gave the equivalence of the equations under both coordinates. These equations are generalized to the n-body central configurations with $(n-2)$-dimension afterward, and one can refer to Chapter 2 in [23] for more details. Especially for $n=3$, we have Heron's formula with $i < j < k$

$$
\mathcal{V}_3 = \mathcal{V}_3(i, j, k) = V_{ijk}
$$

$$
\tag{6}
= \frac{1}{4} \sqrt{-(r_{ij} - r_{ik} - r_{jk})(r_{ij} + r_{ik} - r_{jk})(r_{ij} + r_{ik} + r_{jk})(r_{ij} - r_{ik} + r_{jk})}
= \frac{1}{4} \sqrt{-F_{ijk}(r)} = \frac{1}{4} \sqrt{-F_3(i, j, k)},
$$

where $\mathcal{V}_3 = V_{ijk}$ is the area of the triangle $\triangle P_i P_j P_k$.

We need to point out again that not each vector $r \in (\mathbb{R}^+)^{C_n^2}$ corresponds to a n-body configuration. Here, especially for four bodies, we need the concept of geometrical realizable set observed and raised by Cors and Roberts in [9]:

$$
\tag{7}
\mathcal{G} = \{r \in (\mathbb{R}^+)^6 \mid r_{ij} + r_{jk} \geq r_{ik},\, \forall i, j, k \in\{1,2,3,4\},\, i \neq j \neq k,\, F_4(1,2,3,4) \geq 0\}
$$

Then the normalized configuration space with respect to $r$ is

$$
\tag{8}
\mathcal{N} = \{r \in \mathcal{G} \mid I(r) = I_0,\, F_4(r) = 0\}
$$

The equivalence of the equations of central configurations under both coordinates is guaranteed by the following result from Dziobek (also refer to [28, 29]):

**Proposition 1.** Let $x$ be a planar 4-body configuration, let $r \in \mathcal{N}$ be its corresponding normalized configuration, and let $U|_{\mathcal{N}} : \mathcal{N} \to \mathbb{R}$ be the restriction of the Newtonian potential $U$ to $\mathcal{N}$. Then, $x$ is a planar central configuration if and only if $r$ is a critical point of $U|_{\mathcal{N}}$ with respect to $r$.

This means that $x$ is a planar central configuration if and only if $r$ is a critical point of the Lagrangian function

$$
\tag{9}
\tilde{W}(r) = U(r) + \lambda (I(r) - I_0) + \sigma F_4(r),
$$

with $\lambda > 0$ and $\sigma$ its multipliers.

### 2.2. The reduced Lagrangian function for convex quadrilateral configurations.

Without loss of generality, we fix an order of the four particles such that $P_1, P_2, P_3$ and $P_4$ lie anticlockwise along a strictly convex quadrilateral, see Figure 1. The key is finding a proper constraint for this general convex quadrilateral. Let

$$
\tag{10}
C(r) = 2 r_{12} r_{34} \cos \theta + r_{23}^2 + r_{14}^2 - r_{24}^2 - r_{13}^2 = 0,
$$

where θ denotes the angle between the extensions of the sides r_{12} and r_{34}. Since we can define that θ = 0 if the line P_{1}P_{2} and P_{3}P_{4} are parallel, i.e., a trapezoid shape, hence θ ∈ [0, π/2], see Figure 2. This constraint comes from Josefsson’s paper [16], in which, in Theorem 10, he gave (10) as a necessary condition of a convex quadrilateral. If we denote by

\[
\tilde{C} = \{ r \in \mathcal{G} \mid C(r) = 0 \},
\]

we can easily check that the collinear and the strictly concave configurations are not in this set. However, the 3-body collinear case does, i.e., the non-strictly convex one with a triangular convex hull. Still, fortunately, it cannot be a central configuration according to the Perpendicular Bisector Theorem.

**Proposition 2.** *For a strictly convex quadrilateral configuration \( r \in \mathcal{G} \),*

\[
(11) \qquad F_4(r) = C(r) \cdot A(r),
\]

*where \( A(r) = 32V_{124}V_{234} \neq 0 \).*

*Proof.* Firstly, since \( r \in \mathcal{G} \) is a strictly convex quadrilateral configuration, we have \( C(r) = 0 \). Substituting

\[
(12) \qquad r_{13}^2 = 2r_{12}r_{34} \cos\theta + r_{23}^2 + r_{14}^2 - r_{24}^2
\]

into \( F_4(r) \) with some simplification and using the notations in (6), we have

\[
(13) \qquad F_4(r) = C(r) \cdot A(r) - 2 \left( \left( \frac{A(r)}{4r_{24}} \right)^2 - B(r)^2 \right),
\]

where the second equality of \(A(r)\) is obtained by eliminating \(r^2_{13}\) with (12). In general, we can eliminate any of the four square items in \(C(r)\) to get different versions of the decomposition of \(F_4\). Secondly, we claim that

\[
A(r) = 4r_{24}B(r)
\]

for \(\theta \in [0, \pi/2]\). It suffices to show that

\[
(15) \quad \cos \theta = \frac{(r^2_{12} - r^2_{14} + r^2_{24})(-r^2_{23} + r^2_{24} + r^2_{34})}{4r_{12}r^2_{24}r_{34}} + \frac{16V_{124}V_{234}}{4r_{12}r^2_{24}r_{34}}.
\]

If we denote by \(\alpha = \angle P_1P_2P_4\) and \(\beta = \angle P_2P_4P_3\), then \(\alpha, \beta \in (0, \pi)\) in a strictly convex quadrilateral. One can see in Figure 2 that with the law of cosines and the area formula of triangles in sines, we have

\[
(16) \quad
\begin{cases}
r^2_{12} - r^2_{14} + r^2_{24} = 2r_{12}r_{24} \cos \alpha, \\
-r^2_{23} + r^2_{24} + r^2_{34} = 2r_{24}r_{34} \cos \beta, \\
V_{124} = \frac{r_{12}r_{24} \sin \alpha}{2}, \\
V_{234} = \frac{r_{24}r_{34} \sin \beta}{2}.
\end{cases}
\]

Substituting (16) into the right side of (15) we have

\[
(17) \quad \frac{(r^2_{12} - r^2_{14} + r^2_{24})(-r^2_{23} + r^2_{24} + r^2_{34})}{4r_{12}r^2_{24}r_{34}} + \frac{16V_{124}V_{234}}{4r_{12}r^2_{24}r_{34}} = \cos(\beta - \alpha).
\]

We draw a parallel line of \(P_1P_2\) crossing \(P_4\), and can easily find that \(\theta = \beta - \alpha\), which leads to the equality of (15), see Figure 2. In general, \(\theta = |\beta - \alpha|\) holds, which does not influence the cosine value. The expression of \(A(r)\) is obtained directly. □

Remark 1. In the proof, we derive (11) by replacing \(r^2_{13}\) with the expression in (12). It is almost the same if we eliminate \(r^2_{24}\) instead, with \(A(r)/(2r_{24})\) and \(B(r)\) changing correspondingly. Things will be a little different if we do the same thing to \(r^2_{14}\) or \(r^2_{23}\) since one can easily check that the relationship of the angles becomes \(\theta + \alpha + \beta = \pi\).

\[
(14) \quad
\begin{cases}
A(r) = 2(r^2_{24}(r^2_{13} - r^2_{34}) + r^2_{12}(r^2_{23} - r^2_{24} - r^2_{34}) + r^2_{14}(-r^2_{23} + r^2_{34}) \\
\qquad + 2r_{12}r^2_{24}r_{34} \cos \theta), \\
\qquad = 8r_{12}r^2_{24} r_{34} \left(\cos \theta - \frac{(r^2_{12} - r^2_{14} + r^2_{24})(-r^2_{23} + r^2_{24} + r^2_{34})}{4r_{12}r^2_{24} r_{34}}\right), \\
B(r) = \frac{\sqrt{F_3(1, 2, 4)F_3(2, 3, 4)}}{2r_{24}} = \frac{8V_{124}V_{234}}{r_{24}},
\end{cases}
\]

**Remark 2.** *Here, we mention some special cases of convex quadrilateral configurations.*

angles \((\alpha + \gamma) + (\beta + \psi) = \pi\) and relationships derived from the law of sines in triangles, i.e.,

\[
\frac{r_{12}}{\sin \psi} = \frac{r_{13}}{\sin(\gamma + \alpha)} = \frac{r_{14}}{\sin \alpha} = \frac{r_{23}}{\sin \beta} = \frac{r_{24}}{\sin(\psi + \alpha)} = \frac{r_{34}}{\sin \gamma} = d
\]

to replace all the mutual distances with \(d\) and sines, then we get \(C(r) - 2 \cos \theta P(r) = 0\). Since \(\cos \theta \not\equiv 0\), this implies that \(P(r) = 0\), i.e., the well-known Ptolemy's theorem for this co-circular quadrilateral. In addition, according to Theorem 6 in [30], Ptolemy's theorem also holds for the 4-body collinear case. For an arbitrary quadrilateral, the inequality \(P \geq 0\) holds; see [25].

Let \(\mathcal{M} = \{ r \in \mathbb{R}^6 \mid I(r) - I_0 = 0,\, C(r) = 0 \}\), and denote by
\[
\mathcal{M}^+ = \{ r \in (\mathbb{R}^+)^6 \mid I(r) - I_0 = 0,\, C(r) = 0 \}
\]
the subset of \(\mathcal{M}\) with respect to positive \(r_{ij}\)s. Let
\[
\mathcal{C} = \{ r \in \mathcal{G} \cap \mathcal{M}^+ \mid F_4(r) = 0 \},
\]
which is the collection of configurations we care about. The relationship of these sets is shown in Figure 4. Since the topology of \(\mathcal{C}\) is complicated, we turn to study a larger region \(\mathcal{M}^+\). In what follows, we will see that the Eulerian characteristic of \(\mathcal{M}^+\) is 1.

With the decomposition of \(F_4(r)\) in (11), we have

Lemma 1. For any \(r \in \tilde{\mathcal{C}}\), we have \(\nabla_r F_4(r) = \nabla_r C(r) \cdot A(r)\).

1. Proof. From (11) we have
$$\nabla_r F_4(r) = \nabla_r C(r) \cdot A(r) + C(r) \cdot \nabla_r A(r).$$

2. For any $r \in \tilde{C}$, we have $C(r) = 0$, and the equality in the lemma holds obviously.

3. Then we conclude that

4. **Proposition 3.** *Let $r \in C$. Then $r$ is the critical point of $\tilde{W}(r)$ if and only if it is the critical point of*

$$W(r) = U(r) + \lambda(I(r) - I_0) + \eta C(r),$$

*where* $\lambda$ *and* $\eta = \sigma \cdot A(r)$ *are multipliers.*

5. *Proof. We can derive this result by calculating the critical point equations of $\tilde{W}(r)$ and $W(r)$. *

6. Here, we need to point out that $\eta \neq 0$ if and only if $r \in G$ is strictly convex or only $P_1, P_2$ and $P_3$ are collinear.

7. 2.3. *The type of the critical point of* $W(r)$. *The critical point equations of* $W(r)$ *are*

$$
\begin{cases}
S_{12} + 2\eta \cdot r_{34} \cos\theta = 0, \\
S_{13} - 2\eta \cdot r_{13} = 0, \\
S_{14} + 2\eta \cdot r_{14} = 0, \\
S_{23} + 2\eta \cdot r_{23} = 0, \tag{18}\\
S_{24} - 2\eta \cdot r_{24} = 0, \\
S_{34} + 2\eta \cdot r_{12} \cos\theta = 0,
\end{cases}
$$

where

$$
\left\{
\begin{aligned}
& S_{ij} = m_i m_j r_{ij} (\delta - r_{ij}^{-3}), \quad 1 \le i < j \le 4, \\
& \delta = \frac{\lambda}{m} > 0.
\end{aligned}
\right.
$$

8. We continue to compute the Hessian of $W(r)$ and obtain

$$
D^2 W(r) =
\begin{bmatrix}
R_{12} & 0 & 0 & 0 & 0 & 2 \eta \cos\theta \\
0 & R_{13} - 2\eta & 0 & 0 & 0 & 0 \\
0 & 0 & R_{14} + 2\eta & 0 & 0 & 0 \\
0 & 0 & 0 & R_{23} + 2\eta & 0 & 0 \\
0 & 0 & 0 & 0 & R_{24} - 2\eta & 0 \\
2\eta \cos\theta & 0 & 0 & 0 & 0 & R_{34}
\end{bmatrix},
$$

where

$$
R_{ij} = m_i m_j \left(\delta + \frac{2}{r_{ij}^3}\right) > 0, \quad 1 \leq i < j \leq 4.
$$

9. **Lemma 2.** *Suppose that* $\tilde{r} \in \mathcal{M}^+$ *is a critical point of* $W(r)$. *Then* $\tilde{r}$ *is a non-degenerate local minimum of* $W(r)$.

ON THE UNIQUENESS OF THE STRICTLY CONVEX QUADRILATERAL CENTRAL CONFIGURATION

Proof. We can easily get four positive eigenvalues of the six of \( D^2 W(r) \) using the corresponding expressions of \( 2\eta \) in (18)

\[
\begin{cases}
\zeta_1 = R_{13} - 2\eta = \dfrac{m_1 m_3}{r_{13}^3} > 0, \\
\zeta_2 = R_{14} + 2\eta = \dfrac{m_1 m_4}{r_{14}^3} > 0, \\
\zeta_3 = R_{23} + 2\eta = \dfrac{m_2 m_3}{r_{23}^3} > 0, \\
\zeta_4 = R_{24} - 2\eta = \dfrac{m_2 m_4}{r_{24}^3} > 0.
\end{cases}
\]

The left two \( \zeta_5 \) and \( \zeta_6 \) are the roots of the following quadratic equation of \( \zeta \)
\[
\zeta^2 + a_1 \zeta + a_0 = 0,
\]
where
\[
\begin{cases}
a_1 = -(\zeta_5 + \zeta_6) = - (R_{12} + R_{34}) < 0, \\
a_0 = \zeta_5 \cdot \zeta_6 = 4 r_{12}^3 r_{13}^6 r_{14}^6 r_{23}^6 r_{24}^6 r_{34}^3 \left( R_{12} R_{34} - 4 \eta^2 r_{12}^3 r_{34}^3 \cos^2 \theta \right).
\end{cases}
\]

To show \( \zeta_5 \) and \( \zeta_6 \) are both positive, it suffices to show \( a_0 > 0 \). In fact, from the first and the last equations in (18) and the relationship of \( S_{ij} \) and \( R_{ij} \), we have

\[
\cos \theta = -\frac{S_{12}}{2 \eta r_{34}} = \frac{r_{12}(R_{12} - 3/r_{12}^3)}{2 \eta r_{34}},
\]
\[
= -\frac{S_{34}}{2 \eta r_{12}} = \frac{r_{34}(R_{34} - 3/r_{34}^3)}{2 \eta r_{12}}.
\]

Substituting the above two equalities at the same time into the expression of \( a_0 \) to replace each \( \cos \theta \), we have
\[
a_0 = 3 m_1 m_2 m_3 m_4 \left( 1 + (r_{12}^3 + r_{34}^3) \delta \right) > 0,
\]
which implies that \( \zeta_5, \zeta_6 \) are both positive. From the above analysis, we derive that the critical point of \( W(r) \), if it exists, is a local minimum with Morse index 0. \(\Box\)

## 2.4. The topology of \( \mathcal{M}^+ \).

Lemma 3. \( \chi ( \mathcal{M}^+ ) = 1 \).

Proof. We can only consider the equal mass case since the Eulerian characteristic number is invariant under continuous deformations. Suppose that \( I_0 = \frac{1}{2m} \) and \( m_1 = \cdots = m_4 = 1 \). Let \( \mathcal{M}_0 \) and \( \mathcal{M}_0^+ \) be the sets corresponding to the equal mass cases of \( \mathcal{M} \) and \( \mathcal{M}^+ \), respectively. We denote by \( S^n(\rho) \) the \( n \)-dimensional sphere with radius \( \rho \) and
\[
S^5 ( 1 ) = \left\{ r \in \mathbb{R}^6 \bigl| r_{12}^2 + r_{13}^2 + r_{14}^2 + r_{23}^2 + r_{24}^2 + r_{34}^2 = 1 \right\}.
\]

Then, with \( r_{ij} \geq 0 \), the constraints in \( M_0^+ \) are

\[
\begin{cases}
r_{12}^2 + r_{13}^2 + r_{14}^2 + r_{23}^2 + r_{24}^2 + r_{34}^2 = 1, \\
2r_{12}r_{34} \cos\theta + r_{23}^2 + r_{14}^2 - r_{24}^2 - r_{13}^2 = 0.
\end{cases} \tag{19}
\]

We first assume that \( \cos\theta \neq 0 \). Divide both sides of the second equation by \( \cos\theta \) and then use the first equality to add and subtract it separately. We have

\[
\begin{cases}
(r_{12} + r_{34})^2 + \left( \frac{1}{\cos\theta} + 1 \right) (r_{14}^2 + r_{23}^2) = 1 + \left( \frac{1}{\cos\theta} - 1 \right) (r_{24}^2 + r_{13}^2), \\
(r_{12} - r_{34})^2 + \left( \frac{1}{\cos\theta} + 1 \right) (r_{24}^2 + r_{13}^2) = 1 + \left( \frac{1}{\cos\theta} - 1 \right) (r_{14}^2 + r_{23}^2).
\end{cases}
\tag{20}
\]

If we denote by
\[
\begin{cases}
v_1 = r_{12} + r_{34} \geq 0, \\
w_1 = r_{12} - r_{34} \leq v_1, \\
\Theta(\theta) = \frac{1}{\cos\theta} + 1 \geq 2, \\
G_1 = r_{14}^2 + r_{23}^2 \geq 0, \\
G_2 = r_{24}^2 + r_{13}^2 \geq G_1,
\end{cases}
\tag{21}
\]

then (20) becomes

\[
\begin{cases}
v_1^2 + \Theta G_1 = 1 + (\Theta - 2)G_2, \\
w_1^2 + \Theta G_2 = 1 + (\Theta - 2)G_1.
\end{cases}
\tag{22}
\]

Furthermore \(0 \leq G_1 \leq \frac{1}{2}\) holds since from the first equation in (19) we have \(G_1 + G_2 \leq 1\). Noticing that \(v_1^2 - w_1^2 = 2(\Theta - 1)(G_2 - G_1) \geq 0\) implies \( |w_1| \leq |v_1| \). Then \(w_1^2 = 1 + (\Theta - 2)G_1 - \Theta G_2 \geq 0\) must hold to make sense. This leads to the following inequalities

\[
G_1 \leq G_2 \leq \frac{1+(\Theta-2)G_1}{\Theta},
\]

Since \(G_1 \leq \frac{1+(\Theta-2)G_1}{\Theta} = G_1 + \frac{1-2G_1}{\Theta}\) holds for any \(0 \leq G_1 \leq \frac{1}{2}\). Denote by

\[
D_1 = \left\{ (r_{14}, r_{23}) \in (\mathbb{R}^+)^2 \,\middle|\, 0 \leq r_{14}^2 + r_{23}^2 = G_1 \leq \frac{1}{2} \right\},
\]

then \(D_1\) is contractible in the 2-dimension \((r_{14}, r_{23})\)-plane, i.e., it forms a quarter of a disk with radius \(1/2\). Now we fix a point \((r_{14}, r_{23}) \in D_1\). Then the point \((r_{13}, r_{24})\) satisfying (2.4) forms a contractible region

\[
D_2 = \left\{ (r_{13}, r_{24}) \in (\mathbb{R}^+)^2 \,\middle|\, G_1 \leq r_{13}^2 + r_{24}^2 \leq \frac{1 + (\Theta - 2) G_1}{\Theta} \right\},
\]

i.e., a quarter of a ring belt in the 2-dimension \((r_{13}, r_{24})\)-plane, see Figure 5.

For a fixed point \((r_{13}, r_{24}, r_{14}, r_{23}) \in D\), we find that \(v_1\) and \(w_1\) are determined uniquely by (22). Without loss of generality, we assume that \(w_1 = r_{12} - r_{34} \leq 0\) holds. Hence the projection

\[
\tilde{p}_1 : (r_{13}, r_{24}, r_{14}, r_{23}) \to (r_{14}, r_{23}),
\]

which induces a fibration \(p_1 : D \to D_1\) with \(D_2\) its fiber and \(D_1\) its base. Since both \(D_1\) and \(D_2\) are contractible, \(D\) is contractible.

To summarize the above, we consider a projection

\[
\tilde{p} : (r_{13}, r_{24}, r_{14}, r_{23}, v_1, w_1) \to (r_{13}, r_{24}, r_{14}, r_{23})
\]

induces a fibration \(p : M_0^+ \to D\) with \(D\) its base and \(\{(v_1, w_1)\}\) its fiber, i.e., a point, and it is contractible. Hence, \(M_0^+\) is contractible for \(\cos \theta \in (0, 1]\).

If \(\cos \theta = 0\), we can reduce (19) to
\[
\begin{cases}
r_{14}^2 + r_{23}^2 = \frac{1}{2}(1 - r_{12}^2 - r_{34}^2), \\
r_{13}^2 + r_{24}^2 = \frac{1}{2}(1 - r_{12}^2 - r_{34}^2).
\end{cases}
\]

\[
\mathcal{H} = \{(r_{12}, r_{34}) \in (\mathbb{R}^+)^2 \mid r_{12}^2 + r_{34}^2 \leq 1\}
\]
is contractible in the 2-dimension \((r_{12}, r_{34})\)-plane. Now fix a point \((r_{14}, r_{23}) \in \mathcal{H}\). Then
\[
\mathcal{H}_1 = \left\{(r_{14}, r_{23}) \in (\mathbb{R}^+)^2 \mid r_{14}^2 + r_{23}^2 = \frac{1}{2}(1 - r_{12}^2 - r_{34}^2)\right\}
\]

forms a contractible quarter of a circle with radius
\[
\sqrt{\frac{1}{2}(1 - r_{12}^2 - r_{34}^2)}
\]
in the 2-dimension \((r_{12}, r_{34})\)-plane, and similarly, we have

\[
\mathcal{H}_2 = \left\{ (r_{13}, r_{24}) \in (\mathbb{R}^+)^2 \ \bigg| \ r_{13}^2 + r_{24}^2 = \frac{1}{2}(1 - r_{12}^2 - r_{34}^2) \right\}.
\]

To sum up, the projection \(\tilde{s}_1 : (r_{14}, r_{23}, r_{12}, r_{34}) \to (r_{12}, r_{34})\) induces a fibration
\(s_1 : \tilde{\mathcal{H}} \to \mathcal{H}\), in which both the base \(\mathcal{H}\) and the fiber \(\mathcal{H}_1\) are contractible, so \(\tilde{\mathcal{H}}\) is contractible. The projection \(\tilde{s} : (r_{13}, r_{24}, r_{14}, r_{23}, r_{12}, r_{34}) \to (r_{14}, r_{23}, r_{12}, r_{34})\) induces a fibration \(s : M_0^+ \to \tilde{\mathcal{H}}\), in which both the base \(\tilde{\mathcal{H}}\) and the fiber \(\mathcal{H}_2\) are contractible, so \(M_0^+\) is contractible for \(\cos \theta = 0\). This lead to that \(\chi(M_0^+) = 1\).

The relationship between \(M_0^+\) and \(M^+\) is easy to understand. In fact, \(M_0^+\) is a contractible subset of \(S^5(1)\), which is homeomorphic to the corresponding 5-dimension ellipsoid
\[
\mathcal{E}^5(1) = \left\{ r \in \mathbb{R}^5 \ \bigg| \ \sum_{i<j}^4 m_i m_j r_{ij}^2 = 1 \right\}.
\]
The cone \(C(r) = 0\) passing through the origin intersects \(\mathcal{E}^5(1)\) in \(M^+ \subset \mathcal{E}^5(1)\) via the homeomorphic map. Hence we have \(\chi(M^+) = \chi(M_0^+) = 1\). □

## 2.5. The uniqueness.

**Lemma 4.** *\(U(r)\) has a unique critical point on \(M^+\).*

*Proof.* From Lemma 2, the critical points of \(U(r)\) restricting on \(M^+\) are non-degenerate local minimum with Morse index 0, which implies that \(U\) is a Morse function. Noticing that \(r \to \partial M^+\) leads to \(U \to +\infty\), the minimum point exists. Then, from the Morse equation below where \(\alpha_q\) denotes the number of the non-degenerate critical points with Morse index \(q\)
\[
\sum (-1)^q \alpha_q = \chi(M^+) = 1
\]
we have \(\alpha_0 = 1\) provided \(q = 0\). □

Finally, we have

*The proof of Theorem 1.* From Proposition 3, if \(r\) is a convex quadrilateral central configuration, it is a critical point of the Lagrangian function \(W(r)\), i.e., the potential function \(U\) restricted on \(M^+\). From Lemma 2, this critical point must be a non-degenerate local minimum. Since \(C \subset M^+\), we conclude that \(U\) has at most one critical point restricted on \(C\). □

## ACKNOWLEDGEMENTS

This work is partially supported by the NSF of China (No. 12071316). The authors would like to thank the NSF of China.

All data generated or analyzed during this study are included in this published article.

The authors declared that they have no conflicts of interest in this work.

16                      YANGSHANSHAN LIU AND SHIQING ZHANG

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[System]
You are a **scientific‐rigor auditor**. You will receive the parsed contents of a research paper. Your job is to identify **only** those errors or flaws that directly undermine the **scientific validity** of the paper’s methods, analyses, or conclusions.

Your sole focus is to identify flaws—such as errors in experimental design, data integrity, calculations, statistical inference, or reproducibility—that directly call into question the validity of a specific claim, paragraph, or the paper as a whole.

**Do not** report issues that are purely presentational, rhetorical, stylistic, or related to citation practices.

---

After you’ve done a **detailed walkthrough** of the paper, output exactly in this format—no extra keys or commentary:

```
<analysis>
{detailed walk‑through of how you checked each section/figure and why you flagged (or did not flag) any flaw}
</analysis>

<response>
{
  "has_error": <true|false>,
  "errors": [
    {
      "location": "Section 2.1",
      "description": "Claim that ‘all X are Y’ is contradicted by citation [5], which shows examples of non‑Y X."
    },
    {
      "location": "Figure 3",
      "description": "XAxis labeled ‘Time (s)’ but units appear to be milliseconds; scale bar mismatches caption."
    }
    // …more entries…
  ]
}
</response>
```

- **Do not** include any other keys or prose outside these two tagged blocks.
- **Do not** report stylistic or citation issues.
- Be as precise as possible about **where** (section ID or figure/table) and **what** the scientific flaw is.
- Each description for the errors must be rooted in a scientific rationale explaining why it is 'wrong' (not how it could be improved).

Begin your analysis now.

[User]
**Abstract.** In the general setting of a locally compact Abelian group \( G \), the Delsarte extremal problem asks for the supremum of integrals over the collection of continuous positive definite functions \( f : G \to \mathbb{R} \) satisfying \( f(0) = 1 \) and having \( \operatorname{supp} f_+ \subset \Omega \) for some measurable subset \( \Omega \) of finite measure. In this paper, we consider the question of the existence of an extremal function for the Delsarte extremal problem. In particular, we show that there exists an extremal function for the Delsarte problem when \( \Omega \) is closed, extending previously known existence results to a larger class of functions.

The Delsarte problem arose first in the context of discrete codes. Specifically, in [8], P. Delsarte found an upper bound for the number of code words in codes satisfying some prescribed distance condition. The solution was carried out in terms of Krawtchouk polynomials and the MacWilliams transform. Subsequently, in [9], P. Delsarte, J. M. Goethals, and J. J. Seidel considered another instance of the Delsarte problem in the context of spherical codes in \( \mathbb{R}^d \). In particular, they derived an upper bound for certain spherical codes in terms of coefficients of Gegenbauer expansions. Various instances of the Delsarte problem occur in the context of coding theory, see for instance [18] and the references within. It appears in the context of sphere packing where it is used to obtain upper bounds for the density of sphere packings, see [1,2,4,11,12,14–16,19]. For more on the Delsarte problem and its generalisation to locally compact Abelian (LCA) groups, see [3]. Based on the Poisson summation formula, in [6], H. Cohn and N. Elkies use the Delsarte scheme to show that the sphere packing density is bounded above

Department of Mathematics and Applied Mathematics, University of Cape Town,
Private Bag X1, 7701, Rondebosch, Cape Town, South Africa
e-mail: rmbmit001@myuct.ac.za

(Received March 1, 2024; revised November 18, 2024; accepted November 18, 2024)

by the Delsarte constant computed over a class of continuous positive definite functions \( f : \mathbb{R}^d \to \mathbb{R} \) satisfying Delsarte-type problem restrictions. In [19], M. Viazovska showed that the bound obtained by the Delsarte scheme in \( \mathbb{R}^8 \) is exact by constructing an extremal function for the Delsarte problem in that context. So, the problem of the existence of an extremal function for the Delsarte problem seems interesting. In this direction, in [3, 10] the problem of the existence of an extremal function for a Delsarte-type extremal problem is considered for the Gorbachev class, defined as follows. Let \( G \) be a LCA group. Note that we assume LCA groups to be Hausdorff. Consider

\[
\mathcal{G}_G(\Omega, Q) := \{ f \in P_1(G) \cap L^1(G) : \text{supp } f_+ \subset \Omega, \text{ supp } \hat{f} \subset Q \},
\]

where \( \Omega \) is a fixed closed subset of \( G \) of finite Haar measure, \( Q \) is a fixed compact subset of the dual group \( \widehat{G} \) of \( G \), \( \hat{f} \) is the Fourier transform of \( f \), \( P_1(G) \) is the collection of continuous positive definite functions \( f : G \to \mathbb{R} \) satisfying \( f(0) = 1 \), \( f_+ \) is the positive part of \( f \) given by \( f_+(g) := \max\{ f(g), 0 \} \) for all \( g \in G \), and supp \( f \) is the support of \( f \), given by \( \text{supp } f := \{ g \in G : f(g) \neq 0 \} \).

In [10], it is shown that \( \mathcal{G}_G(\Omega, Q) \) contains an extremal function for the Delsarte constant:

\[
\mathcal{D}_G(\Omega, Q) := \sup_{f \in \mathcal{G}_G(\Omega, Q)} \int_G f(g) \, d\lambda_G(g),
\]

where \( \lambda_G \) is a fixed Haar measure on \( G \). Note that this result was also proved in [3] for \( G = \mathbb{R}^d \). In this paper, we remove the condition that supp \( \hat{f} \subset Q \). More precisely, we consider the function class

\[
\mathcal{G}_G(\Omega) := \{ f \in P_1(G) \cap L^1(G) : \text{supp } f_+ \subset \Omega \},
\]

where \( \Omega \) is a closed subset of \( G \) of finite Haar measure, and we show the existence of an extremal function for the Delsarte constant

\[
\mathcal{D}_G(\Omega) := \sup_{f \in \mathcal{G}_G(\Omega)} \int_G f(g) \, d\lambda_G(g).
\]

It may happen that the class \( \mathcal{G}_G(\Omega) \) is empty. This happens when \( 0 \notin \operatorname{int} \Omega \). This is because if \( f \) is positive definite, then \( |f(g)| \leq f(0) \) for all \( g \in G \), and if \( 0 \notin \operatorname{int} \Omega \), then \( f(0) = 0 \), and hence \( f \) is identically 0. In which case, the class \( \mathcal{G}_G(\Omega) \) is empty because \( f(0) = 1 \) can not hold for a continuous positive definite function \( f \) with \( f_+ \) supported in \( \Omega \). On the other hand, if \( \Omega \) is a neighbourhood of 0, i.e., \( 0 \in \operatorname{int} \Omega \), then there is a symmetric neighbourhood \( U \) of 0 such that \( U + U \subset \Omega \). Then the convolution \( 1_U * 1_U \) of the characteristic function \( 1_U \) of \( U \) with itself is a continuous positive definite function supported in \( \Omega \). We can normalise it to obtain a function that

belongs to \( G_G(\Omega) \). In other words, the class \( G_G(\Omega) \) is nonempty if and only if \( \Omega \) is a neighbourhood of 0.

The reader may note that the condition that \( f \in L^1(G) \) is needed in general for extremal problems in various function classes, such as those considered in [3,15]. However, in this case it follows from \( f \in P_1(G) \) together with \( \lambda_G(\Omega) < \infty \).

## 2. Notation and preliminaries

Let \( G \) be a LCA group with identity 0. Let \( C(G) \) be the space of continuous functions on \( G \) to the complex numbers and put on it the topology of uniform convergence on compact sets. The dual group \( \widehat{G} \) of \( G \) consists of continuous homomorphisms of \( G \) to the multiplicative group \( \mathbb{T} = \{ z \in \mathbb{C} : |z| = 1 \} \).
With respect to pointwise operations and the subspace topology that it inherits as a subset of \( C(G) \), \( \widehat{G} \) is a LCA group. Fix a Haar measure \( \lambda_G \) on \( G \) and let \( L^1(G) \) denote the usual Banach space of integrable complex-valued functions on \( G \). The Fourier transform of \( f \in L^1(G) \) is defined by

\[
\widehat{f}(\chi) := \int_G f(g)\overline{\chi(g)}\, d\lambda_G(g) \quad \text{for all } \chi \in \widehat{G}.
\]

The convolution \( f * g \) of functions \( f, g : G \to \mathbb{C} \) is defined by

\[
f * g(x) := \int_G f(y)g(x-y)\, d\lambda_G(y) \quad \text{for all } x \in G,
\]

whenever the integral exists. A function \( f : G \to \mathbb{C} \) is positive definite if the inequality

\[
\sum_{i=1}^n \sum_{j=1}^n c_i \overline{c_j} f(g_i - g_j) \geq 0
\]

holds for all choices of \( n \in \mathbb{N},\, c_i \in \mathbb{C}, \) and \( g_i \in G \). It is a consequence of Bochner's theorem that the Fourier transform of an integrable continuous positive definite function is nonnegative. Consequently, since \( 0 \leq \widehat{f}(0) = \int_G f(g)\, d\lambda_G(g) \), it follows that the integral of an integrable continuous positive definite function is nonnegative. Let \( P(G) \subset C(G) \) denote the collection of continuous positive definite functions on the group \( G \). Define \( P_1(G) := \{ f \in P(G) : f(0) = 1 \} \). Related to the notion of a positive definite function is that of an integrally positive definite function. A function \( f \in L^\infty(G) \) is integrally positive definite if the inequality

\[
\int_G \int_G f(y-x) h(x) \overline{h(y)}\, d\lambda_G(x)\, d\lambda_G(y) \geq 0
\]

Analysis Mathematica 51, 2025

holds for all continuous compactly supported functions \( h \) on \( G \). According to [17, Theorem 1.7.3] an integrally positive definite function agrees locally almost everywhere with a continuous positive definite function. When \( G \) is \( \sigma \)-compact, locally almost everywhere is the same as almost everywhere. Therefore, on a \( \sigma \)-compact group an integrally positive definite function agrees almost everywhere with a continuous positive definite function. The approach to the main problem of this paper is to reduce the problem to the case of \( \sigma \)-compact LCA groups and then extend the solution to the general case. Recall that a LCA group \( G \) is \( \sigma \)-compact if it can be written as a countable union of compact sets.

In this section, we show the existence of an extremal function for the Delsarte extremal problem. Our approach follows that of [10]. If \( H \) is a subgroup of a LCA group \( G \) and \( \varphi : H \rightarrow \mathbb{R} \) is a function, its trivial extension is the function \( \tilde{\varphi} : G \rightarrow \mathbb{R} \), given by

\[
\tilde{\varphi}(g) :=
\begin{cases}
\varphi(g) & \text{if } g \in H, \\
0 & \text{if } g \in G \setminus H.
\end{cases}
\]

LEMMA 1 (see [10, Lemma 6 and proof of Theorem 2]). Let \( H \) be an open subgroup of a LCA group \( G \), and let \( \Omega \) be a subset of \( H \). If a function \( \varphi : H \rightarrow \mathbb{R} \) is in \( \mathcal{G}_H(\Omega) \), then its trivial extension \( \tilde{\varphi} : G \rightarrow \mathbb{R} \) is in \( \mathcal{G}_G(\Omega) \).

PROOF. Suppose that \( \varphi : H \rightarrow \mathbb{R} \) is in \( \mathcal{G}_H(\Omega) \). In particular, \( \varphi \) is continuous, positive definite, has \( \operatorname{supp} \varphi_+ \subset \Omega \), and satisfies \( \varphi(0) = 1 \). Clearly, its trivial extension has \( \operatorname{supp} \tilde{\varphi}_+ \subset \Omega \), and satisfies \( \tilde{\varphi}(0) = 1 \). To see that \( \tilde{\varphi} \) is continuous, note that \( H \), being an open subgroup, is also closed, and hence its complement \( G \setminus H \) is open. Now, \( \tilde{\varphi} \) is continuous at all \( g \in H \) because it coincides with the continuous function \( \varphi : H \rightarrow \mathbb{R} \) on the open set \( H \). It is continuous at all \( g \in G \setminus H \) because it is identically 0 on the open set \( G \setminus H \). It is positive definite by [13, Theorem 32.43(a)]. □

We shall also make use of the following lemma.

LEMMA 2. Let \( G \) be a LCA group, and let \( (K_n)_{n \in \mathbb{N}} \) be a sequence of \( \sigma \)-compact subsets of \( G \). The group \( H \) generated by the union \( \bigcup_{n \in \mathbb{N}} K_n \) is \( \sigma \)-compact. Furthermore, if at least one of the \( K_n \)'s has non-empty interior, then \( H \) is open.

Lemma 2 is standard and the interested reader may consult the proof of [7, Proposition 1.2.1(c)].

Analysis Mathematica 51, 2025

THEOREM 3. Let G be a LCA group, and Ω a subset of G of finite Haar measure. There is an open σ-compact subgroup H of G containing Ω such that

(1)  \( D_G(\Omega) = D_H(\Omega) \).

PROOF. By outer regularity of the Haar measure on G, there is an open identity neighbourhood U in G containing Ω with \( \lambda_G(U) < \infty \). We claim that U lies in an open σ-compact subgroup. To see this, let K be some σ-compact open subgroup of G. One may always obtain such a K by considering a relatively compact symmetric identity neighbourhood V and observing that the union

\( K := \bigcup_{n=1}^{\infty} nV \)

is an open σ-compact subgroup. Now, note that G is a disjoint union of cosets g + K of K with g running through a set of coset representatives of K. Since the cosets of K are disjoint and cover G, the collection of subsets (g + K) ∩ U of U as g runs through the set of coset representatives of K are disjoint and their union is U. Now, either (g + K) ∩ U = ∅ or \( \lambda_G((g + K) \cap U) > 0 \) because (g + K) ∩ U is open, and every open set is either empty or has strictly positive measure. Hence (g + K) ∩ U ≠ ∅ for at most countably many cosets g + K of K, otherwise this would contradict the finiteness of the Haar measure of U. Clearly, U is contained in the subgroup generated by the union of K and the at most countably many open σ-compact cosets g + K of K that meet U nontrivially. Denote this subgroup by H and observe that it is open and σ-compact by Lemma 2. Equip H with Haar measure \( \lambda_H := \lambda_G|_H \). By construction, Ω ⊂ H.

Let us first consider the case when 0 ∉ int Ω. Then, both \( \mathcal{G}_G(\Omega) \) and \( \mathcal{G}_H(\Omega) \) are empty. Therefore, both sides of (1) are equal to 0, and the theorem is proved in this case. Now assume that 0 ∈ int Ω, so that both \( \mathcal{G}_G(\Omega) \) and \( \mathcal{G}_H(\Omega) \) are nonempty.

First, let \( (f_n)_{n \in \mathbb{N}} \) be a \( D_G(\Omega) \)-extremal sequence in \( \mathcal{G}_G(\Omega) \) such that

\[
\int_G f_n(g) d\lambda_G(g) \geq D_G(\Omega) - \frac{1}{n} \quad \text{for all } n \in \mathbb{N}.
\]

Let \( f_n|_H : H \rightarrow \mathbb{R} \) denote the restriction of \( f_n \) to H. Observe that for all n ∈ ℕ, we have

(2)  \(
\int_G f_n(g) d\lambda_G(g) = \int_H f_n(g) d\lambda_G(g) + \int_{G \setminus H} f_n(g) d\lambda_G(g),
\)

Analysis Mathematica 51, 2025

and since supp(f_n)_+ is contained in H, we have that f_n(g) ≤ 0 for all g ∈ G\H. Therefore,

(3)  ∫_{G\H} f_n(g) dλ_G(g) ≤ 0.

From (2) and (3) it follows that

(4)  ∫_H f_n|_H(g) dλ_H(g) = ∫_H f_n(g) dλ_G(g) ≥ ∫_G f_n(g) dλ_G(g).

Now, by the definition of D_H(Ω) and inequality (4), we have

D_H(Ω) ≥ ∫_H f_n|_H(g) dλ_H(g) ≥ ∫_G f_n(g) dλ_G(g) ≥ D_G(Ω) - 1/n.

Thus D_H(Ω) ≥ D_G(Ω).

Now, let (h_n)_{n∈N} be a D_H(Ω)-extremal sequence in G_H(Ω) such that

∫_H h_n(g) dλ_H(g) ≥ D_H(Ω) - 1/n for all n ∈ ℕ.

By Lemma 1, the sequence (h̃_n)_{n∈N} of trivial extensions is in G_G(Ω), and since h̃_n vanishes outside of H, we have that for all n ∈ ℕ,

D_G(Ω) ≥ ∫_G h̃_n(g) dλ_G(g) = ∫_H h_n(g) dλ_H(g) ≥ D_H(Ω) - 1/n.

Thus, D_G(Ω) ≥ D_H(Ω), and hence D_G(Ω) = D_H(Ω), as required. □

Next, we show the existence of an extremal function for the Delsarte extremal problem on LCA groups. We first obtain a solution in the context of σ-compact groups and later extend it to the general case. The reader should note that the condition that Ω is closed kicks in here. We also give some information on how well the extremal function can be approximated, which needs some preliminary results.

LEMMA 4 [17, Theorem 1.4.3]. Let G be a LCA group and f a continuous positive definite function on G. Let V be a symmetric neighborhood of 0 with compact closure and set χ := 1_V/λ_G(V) where 1_V is the characteristic function of V. Then the inequality

|f(x) - f * χ(x)|^2 ≤ 2f(0)λ_G(V)^{-1} ∫_V (f(0) - Re f(g)) dλ_G(g)

holds for all x ∈ G.

In the following, we adapt [17, Lemma 1.5.1, Theorem 1.5.2] to our context, replacing weak* convergence against \( L^1(G) \) and boundedness in \( L^\infty(G) \) by weak convergence and boundedness in \( L^2(G) \) to prove essentially the same results as in [17, Lemma 1.5.1, Theorem 1.5.2]. The proofs are essentially the same, but we include them here for completeness.

LEMMA 5 (compare [17, Lemma 1.5.1]). Let \( G \) be a LCA group. Let \( k \in L^2(G) \), and let \( (\varphi_n)_{n \in \mathbb{N}} \) be a sequence in \( L^2(G) \) with \( \|\varphi_n\|_{L^2(G)} \leq C \) for all \( n \in \mathbb{N} \). If the sequence \( (\varphi_n)_{n \in \mathbb{N}} \) converges weakly in \( L^2(G) \) to a function \( \varphi \in L^2(G) \), then the sequence \( (k * \varphi_n)_{n \in \mathbb{N}} \) converges to \( k * \varphi \) uniformly on the compact subsets of \( G \).

PROOF. For \( h \in L^2(G) \), define
\[
L(h) := \int_G h(g)\varphi(-g) \, d\lambda_G(g), \quad L_n(h) := \int_G h(g)\varphi_n(-g) \, d\lambda_G(g).
\]
The functionals \( L \) and \( L_n \) are continuous with respect to the norm topology on \( L^2(G) \). By Hölder's inequality, \( |L_n(h)| \leq C\|h\|_{L^2(G)} \) for all \( n \in \mathbb{N} \). Therefore, by the Uniform Boundedness Principle, the family \( \{L_n : n \in \mathbb{N}\} \) is equicontinuous on \( L^2(G) \). Note that weak convergence of \( (\varphi_n)_{n \in \mathbb{N}} \) to \( \varphi \) is equivalent to pointwise convergence of \( L_n \) to \( L \), and that pointwise convergence together with equicontinuity of \( (L_n)_{n \in \mathbb{N}} \) implies that \( (L_n)_{n \in \mathbb{N}} \) converges to \( L \) uniformly on the compact subsets of \( L^2(G) \). Let \( K \) be a compact subset of \( G \). The mapping \( G \ni x \mapsto {}_x k \in L^2(G) \), where \( {}_x k \) denotes the left translate of \( k \) by an element \( x \in G \) and given by \( {}_x k(g) = k(x + g) \) for all \( g \in G \), is continuous and hence the set \( \{{}_x k : x \in K\} \) is compact in the norm topology of \( L^2(G) \). The uniform convergence on \( K \) of \( (k * \varphi_n)_{n \in \mathbb{N}} \) now follows from the equations
\[
k * \varphi_n(x) = \int_G k(x + g)\varphi_n(-g) \, d\lambda_G(g) = L_n({}_x k)
\]
and
\[
k * \varphi(x) = L({}_x k). \quad \Box
\]

THEOREM 6 (compare [17, Theorem 1.5.2]). Let \( G \) be a LCA group. If a sequence \( (f_n)_{n \in \mathbb{N}} \) of continuous positive definite functions on \( G \) converges weakly in \( L^2(G) \) to a continuous positive definite function \( f \) and \( \lim_{n \to \infty} f_n(0) = f(0) \), then this sequence converges to \( f \) uniformly on the compact subsets of \( G \).

PROOF. Let \( K \) be a fixed compact subset of \( G \). For any function \( h \) on \( G \), write \( \|h\|_K := \sup\{|h(g)| : g \in K\} \) and \( \|h\|_\infty := \sup\{|h(g)| : g \in G\} \). We show that
\[
\lim_{n \to \infty} \|f_n - f\|_K = 0.
\]

Let V and χ be as in Lemma 4. We have
\[
\|f_n - f\|_K \leq \|f_n - f_n * \chi\|_\infty + \|f_n * \chi - f * \chi\|_K + \|f * \chi - f\|_\infty.
\]

Now let ε > 0 be arbitrary. By the continuity of f, we can choose V small enough so that

\[
2f(0)\lambda_G(V)^{-1} \int_V (f(0) - \operatorname{Re} f(g))\,d\lambda_G(g) < \frac{\varepsilon^2}{9}.
\]

From Lemma 4, we have

\[
\|f * \chi - f\|_\infty < \frac{\varepsilon}{3}.
\]

Since (f_n)_{n\in\mathbb{N}} converges to f weakly in L^2(G) and \(\lim_{n\to\infty} f_n(0) = f(0)\), there exists an N > 0 such that for all \(n \in \mathbb{N}\) with \(n > N\) we have

\[
2f_n(0)\lambda_G(V)^{-1} \int_V (f_n(0) - \operatorname{Re} f_n(g))\,d\lambda_G(g) < \frac{\varepsilon^2}{9}.
\]

Applying Lemma 4 again, we get

\[
\|f_n - f_n * \chi\|_\infty < \frac{\varepsilon}{3}.
\]

By Lemma 5, there exists M > N such that for all \(n \in \mathbb{N}\) with \(n > M\) we have

\[
\|f_n * \chi - f * \chi\|_K < \frac{\varepsilon}{3}.
\]

Altogether, we conclude that \(\|f_n - f\|_K < \varepsilon\) for all \(n > M\), showing that \((f_n)_{n\in\mathbb{N}}\) converges uniformly on the compact subset K. □

We present a version of Mazur's lemma. For a subset A of a Banach space E, we denote by conv(A) the convex hull of A.

## LEMMA 7 (Mazur's lemma, see [5, Corollary 3.8 and Exercise 3.4]).
Let E be a Banach space and let \((x_n)_{n\in\mathbb{N}}\) be a sequence in E converging to \(x \in E\) weakly. Then there exists a sequence \((y_n)_{n\in\mathbb{N}}\) in E such that

\[
y_n \in \operatorname{conv}\left(\bigcup_{i=n}^{\infty} \{x_i\}\right)\quad \text{for all } n \in \mathbb{N},
\]
and \((y_n)_{n\in\mathbb{N}}\) converges strongly to x.

## THEOREM 8.
Let G be a σ-compact LCA group and Ω a closed neighbourhood of 0 with finite Haar measure. Then there exists an extremal function for \(D_G(\Omega)\). Moreover, the extremal function can be approximated uniformly on the compact subsets of G by a \(D_G(\Omega)\)-extremal sequence.

Analysis Mathematica 51, 2025

Here is the accurately regenerated text:

---
PROOF. Let $(f_n)_{n \in \mathbb{N}}$ be a $D_G(\Omega)$-extremal sequence in $\mathcal{G}_G(\Omega)$ chosen such that

\[
\int_G f_n(g)\, d\lambda_G(g) \geq D_G(\Omega) - \frac{1}{n} \quad \text{for all } n \in \mathbb{N}.
\]

Since $\mathcal{G}_G(\Omega)$ is contained in a closed, bounded, and hence a weakly sequentially compact subset of $L^2(G)$, the sequence $(f_n)_{n \in \mathbb{N}}$ has a subsequence converging weakly in $L^2(G)$ to some $f \in L^2(G)$. By replacing $(f_n)_{n \in \mathbb{N}}$ by this subsequence, we assume that $(f_n)_{n \in \mathbb{N}}$ converges weakly in $L^2(G)$ to $f$. Since $\mathcal{G}_G(\Omega)$ is a convex subset of $L^2(G)$, Lemma 7 implies that there is a sequence $(h_n)_{n \in \mathbb{N}}$ in $\mathcal{G}_G(\Omega)$ chosen such that $h_n \in \mathrm{conv}(\bigcup_{i=n}^{\infty} f_i)$ for all $n \in \mathbb{N}$, and converging to $f$ strongly in $L^2(G)$. We claim that the latter sequence is $D_G(\Omega)$-extremal. First note that by the extremality of $(f_n)_{n \in \mathbb{N}}$, we have that for any $\varepsilon>0$ there exists an $N \in \mathbb{N}$ such that for all $n > N$ it holds that

\[
\left| \int_G f_n(g) d\lambda_G(g) - D_G(\Omega) \right| < \varepsilon.
\]

Now, write $h_n = \sum_{k=n}^{\infty} c_k^n f_k$, where $c_k^n \geq 0$, $\sum_{k=n}^{\infty} c_k^n = 1$, and $c_k^n = 0$ for all but finitely many $k \geq n$. Then for any $n > N$, we have

\[
\left| \int_G h_n(g) d\lambda_G(g) - D_G(\Omega) \right| \leq \sum_{k=n}^{\infty} c_k^n \left| \int_G f_k(g) d\lambda_G(g) - D_G(\Omega) \right| < \sum_{k=n}^{\infty} c_k^n \varepsilon = \varepsilon.
\]

Therefore $(h_n)_{n \in \mathbb{N}}$ is $D_G(\Omega)$-extremal. So, replacing, if necessary, the sequence $(f_n)_{n \in \mathbb{N}}$ by this new sequence $(h_n)_{n \in \mathbb{N}}$, assume that $(f_n)_{n \in \mathbb{N}}$ converges to $f$ strongly in $L^2(G)$. By passing to a pointwise almost everywhere convergent subsequence, assume that $(f_n)_{n \in \mathbb{N}}$ converges to $f$ pointwise almost everywhere. The fact that $|f_n(g)| \leq 1$ for all $n \in \mathbb{N}$, $g \in G$, and that $(f_n)_{n \in \mathbb{N}}$ converges almost everywhere to $f$ implies that $f$ is bounded in $L^{\infty}(G)$ by 1.

Since $(f_n)_{n \in \mathbb{N}}$ converges weakly in $L^2(G)$ to $f$, it follows that $f$ is integrally positive definite. The function $f$ being integrally positive definite and $G$ being $\sigma$-compact implies that $f$ agrees almost everywhere with a continuous positive definite function. Thus, by correcting $f$ on a set of Haar measure zero, thus not changing the value of its integral, assume that $f$ is a continuous positive definite function. Continuity of $f$ and the fact that $f$ is a pointwise almost everywhere limit of the functions $f_n$, each of which satisfying $\operatorname{supp}(f_n)_+ \subset \Omega$, implies that $\operatorname{supp} f_+ \subset \Omega$. Since $\operatorname{supp} f_+ \subset \Omega$ we have

\[
(5) \quad \int_G f_+(g) d\lambda_G(g) = \int_{\Omega} f_+(g) d\lambda_G(g) \leq \lambda_G(\Omega) < \infty.
\]

On the other hand, by Fatou's lemma

(6)  \[\int_G f_-(g)\,d\lambda_G(g) \leq \liminf_{n\to\infty} \int_G (f_n)_-(g)\,d\lambda_G(g)\]
    \[\leq \liminf_{n\to\infty} \int_G (f_n)_+(g)\,d\lambda_G(g) \leq \lambda_G(\Omega) < \infty,\]

where the second inequality of (6) is obtained using that \(f_n\) being positive definite and integrable implies that

\[
0 \leq \int_G f_n(g)\, d\lambda_G(g) = \int_G (f_n)_+(g)\, d\lambda_G(g) - \int_G (f_n)_-(g)\, d\lambda_G(g).
\]

Together, (5) and (6) imply that

\[
\int_G |f(g)|\, d\lambda_G(g) = \int_G f_+(g)\, d\lambda_G(g) + \int_G f_-(g)\, d\lambda_G(g) < \infty.
\]

Therefore, \(f \in L^1(G)\). Next, we show that \(f \in \mathcal{G}_G(\Omega)\) and that it is extremal. To that end, note that the sequence \(\big((f_n)_+\big)_{n\in\mathbb{N}}\) of the positive parts of the \(f_n\)'s converges pointwise almost everywhere to \(f_+\), and that \((f_n)_+ \leq 1_\Omega\). Therefore, by Lebesgue's dominated convergence theorem and \(\lambda_G(\Omega) < \infty\), we have

(7)  \[\int_G f_+(g)\,d\lambda_G(g) = \lim_{n\to\infty} \int_G (f_n)_+(g)\,d\lambda_G(g).\]

Recall that from (6) we have the inequality

(8)  \[\int_G f_-(g)\,d\lambda_G(g) \leq \liminf_{n\to\infty} \int_G (f_n)_-(g)\,d\lambda_G(g).\]

Now, subtracting (8) from (7) we have

\[
\int_G f(g)\,d\lambda_G(g) \geq \lim_{n\to\infty} \int_G (f_n)_+(g)\,d\lambda_G(g)
- \liminf_{n\to\infty} \int_G (f_n)_-(g)\,d\lambda_G(g)
\]

\[
= \limsup_{n\to\infty} \int_G \big((f_n)_+(g) - (f_n)_-(g)\big)\,d\lambda_G(g)
\]

\[
= \lim_{n\to\infty} \int_G f_n(g)\,d\lambda_G(g) = D_G(\Omega).
\]

Since \(f\) is continuous, positive definite, integrable, and \(\mathrm{supp}\,f_+ \subset \Omega\) have already been established, to have that \(f \in \mathcal{G}_G(\Omega)\) we only need to show that \(f(0) = 1\). Since

\[
\int_G f(g)\, d\lambda_G(\Omega) \geq D_G(\Omega) > 0,
\]

Analysis Mathematica 51, 2025

f is not the constantly zero function. Therefore, f(0) ≠ 0. If 0 < f(0) < 1, then the function h := f / f(0) is in 𝒢_G(Ω) and has

∫_G h(g) dλ_G(g) = (1 / f(0)) ∫_G f(g) dλ_G(Ω) ≥ (D_G(Ω) / f(0)) > D_G(Ω),

contradicting the extremality of D_G(Ω). Therefore, f(0) = 1 and hence f ∈ 𝒢_G(Ω). Note that this also shows that f is extremal. Finally, since (f_n)_{n∈ℕ} converges to f weakly in L²(G) and lim_{n→∞} f_n(0) = f(0), Theorem 6 implies that (f_n)_{n∈ℕ} converges to f uniformly on the compact subsets of G. □

Now, having secured the existence of an extremal function for σ-compact groups, we extend the solution to general LCA groups to prove our main result.

THEOREM 9. Let G be a LCA group and Ω a closed neighbourhood of 0 with finite Haar measure. There exists an extremal function for D_G(Ω). Moreover, the extremal function can be approximated uniformly on the compact subsets of G by a D_G(Ω)-extremal sequence.

PROOF. On account of Theorem 3, let H be an open σ-compact subgroup of G containing Ω such that D_G(Ω) = D_H(Ω). By Theorem 8, there exists an extremal function f ∈ 𝒢_H(Ω) such that

∫_H f(g) dλ_H(g) = D_H(Ω).

Let f̃ be the trivial extension of f. By Lemma 1, f̃ ∈ 𝒢_G(Ω). We claim that f̃ is an extremal function for D_G(Ω). Indeed,

∫_G f̃(g) dλ_G(g) = ∫_H f(g) dλ_H(g) = D_H(Ω) = D_G(Ω).

As in Theorem 8, (f_n)_{n∈ℕ} converges to f uniformly on the compact subsets of H, and hence (f̃_n)_{n∈ℕ} converges to f̃ uniformly on the compact subsets of G. □

Acknowledgements. The author would like to sincerely thank Professors Elena E. Berdysheva and Szilárd Gy. Révész for their invaluable supervision and encouragement throughout this project. The author is also thankful to the anonymous reviewer for the feedback that helped improve the paper. Finally, the author would also like to thank the Shuttleworth Postgraduate Scholarship of the University of Cape Town for the financial support that made this research possible.

Funding Open access funding provided by University of Cape Town

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