Uniqueness theorem for locally antipodal Delaunay sets (Q511453)

A set \(X\subset{\mathbb R}^d\) is called a \textit{Delaunay set} (with parameters \(r,R>0\)) if it satisfies the following conditions: (1) \(\#\bigl(B_x^{\circ}(r)\cap X\bigr)\leq 1\); (2) \(\#\bigl(B_x(R)\cap X\bigr)\geq 1\). Here, \(B_x^{\circ}(r)\) represents the interior of the (closed) \(d\)-dimensional ball \(B_x(r)\) centered at \(x\) with radius \(r\). Moreover, for \(x\in X\), \(C_x(R)=X\cap B_x(R)\) is the \textit{\(R\)-cluster} of the point \(x\). Finally, a Delaunay set \(X\) is said to be \textit{locally antipodal} if for every \(x\in X\), the \(2R\)-cluster \(C_x(2R)\) is centrally symmetric with respect to \(x\). In the paper under review the authors prove a uniqueness result for locally antipodal Delaunay sets with a given \(2R\)-cluster. More precisely, they show that if \(X\) is a locally antipodal Delaunay set with parameter \(R\) and \(Y\subset{\mathbb R}^d\) is an arbitrary set satisfying that (i) for every \(y\in Y\) the \(2R\)-cluster \(C_y'(2R)\) (in \(Y\)) is antipodal with respect to \(y\) and (ii) \(C_x(2R)=C_x'(2R)\subset X\cap Y\) for \(x\in X\cap Y\), then \(X=Y\).