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 \(n\ge 2\).

从最低温度下费米-狄拉克粒子系统的平衡状态出发，证明了欧几里得球的一个新特征：紧集\(K\subset {{{\mathbb {R}}}^n}\)（至少有两个元素）是一个n维欧几里得球当且仅当对于每对\(x, y\in \partial K\)和每个\(\sigma \in {{{\mathbb {S}} }^{n-1}}\)，\(\frac{1}{2}(x+y)+\frac{1}{2}|xy|\sigma \in K\)或\(\ frac{1}{2}(x+y)-\frac{1}{2}|xy|\sigma \in K\)。作为一种应用，还证明了欧几里得球的这种表征的测量版本，因此先前证明的费米-狄拉克粒子系统平衡态分类的\(n=3\)结果也适用于所有\ (n\ge 2\)。