A characterization of the Euclidean ball via antipodal points

Abstract

Arising from an equilibrium state of a Fermi–Dirac particle system at the lowest temperature, a new characterization of the Euclidean ball is proved: a compact set $K\subset {{{\mathbb {R}}}^n}$ (having at least two elements) is an n-dimensional Euclidean ball if and only if for every pair $x, y\in \partial K$ and every $\sigma \in {{{\mathbb {S}}}^{n-1}}$, either $\frac{1}{2}(x+y)+\frac{1}{2}|x-y|\sigma \in K$ or $\frac{1}{2}(x+y)-\frac{1}{2}|x-y|\sigma \in K$. As an application, a measure version of this characterization of the Euclidean ball is also proved and thus the previous result proved for $n=3$ on the classification of equilibrium states of a Fermi–Dirac particle system holds also true for all $n\ge 2$.