What are the k-means algorithm assumptions? I'm trying to understand what are the assumptions/hypothesis underlying the k-means clustering algorythm; specifically, I'm looking for a research/academic paper listing such hypothesis and explaining the ratio behind them.
Browsing on the internet, I found the following ones:

*

*spherical clusters: all within cluster data are disposed of the center of the cluster;

*cluster size: the size of the clusters in the sample is similar among the clusters identified;

Can you provide some reference about?
Moreover, once you provided that, it would be nice to have the methodology steps to test for such assumptions.
Thanks all in advance!!

There are a bunch of different algorithms that can be applied to the k-means inference problem, so the particular assumptions and statistical interpretation will depend on the specific algorithm you're using.  Notwithstanding this generality, I'm going to assume that you're talking about the "standard" k-means algorithm that proceeds by minimising the within-cluster sum-of-squares for pre-specified values of $k$, where clusters are determined by distance from a set of cluster-means.
Algorithms for this variant of the problem take in a data vector $\mathbf{x}=(x_1,...,x_n)$ and determine a set of means $\boldsymbol{\mu}=\{\mu_1,...,\mu_k\}$ that solve the following optimisation problem:
$$\underset{\boldsymbol{\mu}}{\text{Minimise}} \quad \quad \sum_{i=1}^n \underset{r}{\min} ||x_i - \mu_r||^2.$$
In this optimisation problem the clusters are determined by closeness to the means (each data point is assigned to the cluster for the mean that is closest to it in terms of Euclidean distance) and we minimise the within-cluster sum-of-squares of the resulting clusters.

Statistical Interpretation: Because this optimisation minimises sums-of-squares, the standard statistical interpretation is that it is equivalent to taking the maximum likelihood estimator (MLE) for a statistical model that is a "mixture" of $k$ Gaussian distributions with the same variances but with means given by $\boldsymbol{\mu}$, where each data point is taken to belong to the Gaussian distribution with the closest mean.  In this interpretation, we reframe the obective function as:
$$\boldsymbol{\mu}_\text{MLE} \equiv \underset{\boldsymbol{\mu}}{\text{arg min}} \ \ell_\mathbf{x}(\boldsymbol{\mu})
\quad \quad \quad
\ell_\mathbf{x}(\boldsymbol{\mu}) = \sum_{i=1}^n \log \text{N}(x_i | \mu_{R(i)}, \sigma^2),$$
where the index $R(i) \equiv \text{arg min}_r ||x_i - \mu_r||^2$ gives the group with the closest mean.  Note that this is just one statistical interpretation for one algorithm for k-means — it operates by reinterpreting minimisation of a sum-of-squares as maximisation of a normal density.  There are other interpretations that can be given for other variants of the algorithm; since there are many different algorithms there are many different statistical interpretations and the general problem is complex.
