An extension of the isoperimetric inequality on the sphere (Q750907)

For a subset A of the n-sphere \(S^ n\) in \({\mathbb{R}}^{n+1}\) the author defines the k-diameter of A to be the supremum of the \(\min \{d(x_ i,x_ j)|\) \(1\leq i<j\leq k\}\) where the supremum is taken over every set of k points \(x_ 1,...,x_ k\) of A. Here d(\(\cdot,\cdot)\) denotes the spherical distance on \(S^ n\). The 2-diameter of A is then the ordinary diameter of A. He shows that if \(C\subset S^ n\) is a spherical cap having the same Lebesgue measure as A then the k-diameter of A is not less than the k-diameter of the cap C. The proof follows Benyamini's [\textit{Y. Benyamini}, Longhorn Notes, Univ. Texas, Texas Funct. Anal. Semin., 53-76 (1983-84)] proof of the classical isoperimetric inequality on the sphere using compression operators.