Vol. 44, No. 1, 1973

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The index of convexity and the visibility function

Gerald A. Beer

Vol. 44 (1973), No. 1, 59–67
Abstract

If the integral of the visibility function for a set E is normalized, one arrives at the Index of convexity of E, a measure of the relative convexity of E in terms of the average “area seen” by a variable point of E. As the visibility function is upper semicontinuous on a compact set in En, the Index is upper semicontinuous on the class of all compact sets in En with an appropriate metric. We also investigate natural generalizations of convex and starshaped sets in terms of the visibility function.

Mathematical Subject Classification 2000
Primary: 52A35
Milestones
Received: 14 September 1971
Revised: 28 July 1972
Published: 1 January 1973
Authors
Gerald A. Beer