The notion of point of local
nonconvexity has been an important tool in the study of the geometry of nonconvex
sets, since Tietze characterized, more than fifty years ago, the convex subsets of En
as those connected sets without points of local nonconvexity. It is proved here that
for each convex component K of a closed connected set S in a locally convex space
there exist points of local nonconvexity of S arbitrarily close to K, unless S itself be
convex. Klee’s generalization of the just quoted Tietze’s theorem follows immediately.
The notion of “higher visibility” is introduced in the last section, and three
Krasnosselsky-type theorems involving the points of local nonconvexity are
proved.
