Quantization coefficients for uniform distributions on the boundaries of regular polygons (Q2244502)

Let \(\mu\) be a Borel probability measure on the \(d\)-dimensional Euclidean space \(\mathbb R^d\) equipped with the Euclidean norm \(\|\cdot\|\). Denote by \(V_n(\mu)\) the \(n\)-th quantization error for \(\mu\) defined as \[ V_n(\mu):=\inf\left\{\int_{\mathbb R^d}\min_{a\in A}\|x-a\|\, \mu(\mathrm d x):A\subset\mathbb R^d, 1\leq\mathrm{card}(A)\leq n\right\}. \] Let \(\mu_m\) be the uniform distribution on the boundary of a regular \(m\)-sided polygon inscribed in the unit circle in \(\mathbb R^2\). The authors show that \[ \lim_{n\to\infty} n^2 V_n(\mu_m)=\frac13 m^2\sin^2\left(\frac{\pi}{m}\right), \] thus calculating the one-dimensional quantization coefficient for \(\mu_m\). This result readily implies the conjecture from [\textit{G. Pena} et al., J. Optim. Theory Appl. 188, No. 1, 113--142 (2021; Zbl 1480.60036)] which states that the quantization coefficients for \(\mu_m\) form an increasing sequence which converges to the quantization coefficient for the uniform distribution on the unit circle which is known to be \(\pi^2/3\).