On circumcenter mappings induced by nonexpansive operators

We introduce the circumcenter mapping induced by a set of (usually nonexpansive) operators. One prominent example of a circumcenter mapping is the celebrated Douglas–Rachford splitting operator. Our study is motivated by the Circumcentered–Douglas–Rachford method recently introduced by Behling, Bello Cruz, and Santos in order to accelerate the Douglas–Rachford method for solving certain classes of feasibility problems. We systematically explore the properness of the circumcenter mapping induced by reflectors or projectors. Numerous examples are presented. We also present a version of Browder’s demiclosedness principle for circumcenter mappings.