  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.
