Maximal triangles in space-form balls and simplices in Euclidean balls with prescribed vertex centroid

Abstract

We determine maximum-volume simplices with prescribed interior vertex centroid in Euclidean balls and maximum-area triangles in geodesic balls of simply connected space forms. In full Euclidean dimension at least two, the maximizers are kites: a regular (n−1)-simplex base with an equidistant apex. We prove global optimality and classify equality. The triangle results cover every ambient dimension at least two and every constant curvature, with spherical balls contained in an open hemisphere. The centroid is the arithmetic mean in Euclidean space and the normalized model-vector sum otherwise. Interior vertices are allowed; every equal-weight maximizer lies on the boundary. A change of maximizing triangle shape occurs only in positive curvature, where the transition family has constant area. For prescribed positive vertex masses in Euclidean positive codimension, we characterize attainment of the classical moment bound in a fixed ball and classify equality. The proofs use boundary variations, stationary configurations and Gram determinant inequalities. MSC 2020. Primary 52A40; Secondary 52A55, 51M25. The accompanying public package provides the article source and supplementary mathematical material, with data and code for reproducing the computations.