Vol. 101, No. 1, 1982

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The points of local nonconvexity of starshaped sets

Fausto A. Toranzos

Vol. 101 (1982), No. 1, 209–213

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.

Mathematical Subject Classification 2000
Primary: 52A30
Secondary: 52A35
Received: 3 September 1980
Revised: 10 June 1981
Published: 1 July 1982
Fausto A. Toranzos