A circle of length $10$ is inscribed in a convex polygon with perimeter $15$. What part of the area of this polygon is occupied by the resulting circle?
1. **Given Data and Definitions:**
   - The length (circumference) of the inscribed circle is \(10\).
   - The perimeter of the convex polygon is \(15\).
   - We need to find the fraction of the area of the polygon that is occupied by the circle.

2. **Relationship Between Areas and Perimeters:**
   - We need to prove that:
     \[
     \frac{\text{Area of the circle}}{\text{Circumference of the circle}} = \frac{\text{Area of the polygon}}{\text{Perimeter of the polygon}}
     \]

3. **Area and Circumference of the Circle:**
   - Let \(r\) be the radius of the circle.
   - The circumference of the circle is given by \(2\pi r = 10\), so:
     \[
     r = \frac{10}{2\pi} = \frac{5}{\pi}
     \]
   - The area of the circle is:
     \[
     \text{Area of the circle} = \pi r^2 = \pi \left(\frac{5}{\pi}\right)^2 = \pi \cdot \frac{25}{\pi^2} = \frac{25}{\pi}
     \]

4. **Area and Perimeter of the Polygon:**
   - Let’s denote the area of the polygon as \(A_{\text{polygon}}\) and its perimeter as \(P_{\text{polygon}} = 15\).

5. **Proving the Relationship:**
   - Consider a circular sector of central angle \(\alpha\) between two successive points of tangency.
   - The area of the sector is:
     \[
     \text{Area of the sector} = \frac{1}{2} \alpha r^2
     \]
   - The length of the arc is:
     \[
     \text{Length of the arc} = \alpha r
     \]
   - Therefore, the ratio of the area of the sector to the length of the arc is:
     \[
     \frac{\text{Area of the sector}}{\text{Length of the arc}} = \frac{\frac{1}{2} \alpha r^2}{\alpha r} = \frac{r}{2}
     \]

6. **Corresponding Part of the Polygon (a Kite):**
   - For the corresponding part of the polygon (a kite), the area is:
     \[
     \text{Area of the kite} = 2 \cdot \frac{1}{2} r \cdot r \cdot \tan\left(\frac{\alpha}{2}\right) = r^2 \tan\left(\frac{\alpha}{2}\right)
     \]
   - The length of the kite is:
     \[
     \text{Length of the kite} = 2r \tan\left(\frac{\alpha}{2}\right)
     \]
   - Therefore, the ratio of the area of the kite to its length is:
     \[
     \frac{\text{Area of the kite}}{\text{Length of the kite}} = \frac{r^2 \tan\left(\frac{\alpha}{2}\right)}{2r \tan\left(\frac{\alpha}{2}\right)} = \frac{r}{2}
     \]

7. **Summing Up the Ratios:**
   - Let \(a_k\) be the area of the \(k\)-th sector and \(b_k\) be the length of the \(k\)-th arc.
   - We have:
     \[
     \frac{\sum_{k} a_k}{\sum_{k} b_k} = \frac{\frac{r}{2} \sum_{k} b_k}{\sum_{k} b_k} = \frac{r}{2}
     \]
   - This is valid for the union of circular sectors as well as for the union of kites.

8. **Final Calculation:**
   - Using the given perimeters and areas:
     \[
     \frac{\text{Area of the circle}}{\text{Circumference of the circle}} = \frac{\frac{25}{\pi}}{10} = \frac{25}{10\pi} = \frac{5}{2\pi}
     \]
     \[
     \frac{\text{Area of the polygon}}{\text{Perimeter of the polygon}} = \frac{A_{\text{polygon}}}{15}
     \]
   - Equating the two ratios:
     \[
     \frac{5}{2\pi} = \frac{A_{\text{polygon}}}{15}
     \]
     \[
     A_{\text{polygon}} = 15 \cdot \frac{5}{2\pi} = \frac{75}{2\pi}
     \]

9. **Fraction of the Area:**
   - The fraction of the area of the polygon occupied by the circle is:
     \[
     \frac{\text{Area of the circle}}{\text{Area of the polygon}} = \frac{\frac{25}{\pi}}{\frac{75}{2\pi}} = \frac{25}{\pi} \cdot \frac{2\pi}{75} = \frac{50}{75} = \frac{2}{3}
     \]

The final answer is \(\boxed{\frac{2}{3}}\).