A characterization of the Euclidean ball via antipodal points

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 \(n\ge 2\).