On circumcenters of finite sets in Hilbert spaces

A well-known object in classical Euclidean geometry is the circumcenter of a triangle, i.e., the point that is equidistant from all vertices. The purpose of this paper is to provide a systematic study of the circumcenter of sets containing finitely many points in Hilbert space. This is motivated by recent works of Behling, Bello Cruz, and Santos on accelerated versions of the Douglas--Rachford method. We present basic results and properties of the circumcenter. Several examples are provided to illustrate the tightness of various assumptions.