Strong negative type in spheres (Q2206211)

Let \((X,d)\) be a metric space. This metric space is said to have \textit{negative type}, if for all \(n\geq 1\), all \(x_i\in X\) and \(\alpha_i\in \mathbb{R}\) for \(i=1,2,\dots,n\), the inequality \(\sum_{i=1}^n\sum_{j=1}^n \alpha_i\alpha_j d(x_i,x_j) \leq 0\). The metric space is said to have \textit{strictly negative type}, if equality holds in this inequality if and only if every \(\alpha_i\) equals zero. It is known that all Euclidean spaces have strictly negative type. A (Borel) probability measure \(\mu\) on \(X\) is said to have a \textit{finite first moment}, if for some (and hence for all) \(o\in X\), \(\int d(o,x) d\mu(x)<\infty\). Suppose that \(\mu_1,\mu_2\) have finite first moments. It can be shown that \(\int d(x_1,x_2) d(\mu_1-\mu_2)^2(x_1,x_2)\leq 0\). \((X,d)\) is said to have \textit{strong negative type}, if in this inequality equality holds if and only if \(\mu_1=\mu_2\). It is known that spheres have negative type, but only subsets with at most one pair of antipodal points have strict negative type. The paper under review shows that subsets of spheres with at most one pair of antipodal points have strong negative type. This implies that the function of expected distances to points determines uniquely the probability measure on such a set.