Vol. 53, No. 2, 1974

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The index of convexity and parallel bodies

Gerald A. Beer

Vol. 53 (1974), No. 2, 337–345

Intuitively, the visibility function for a set C in Rn measures the n-dimensional volume of the star of a variable point of C. Suppose that the visibility function for C is measurable. If the measure of C is positive, normalizing the integral of the function produces a measure of the relative convexity of C, called the Index of convexity of C. The purpose of this paper is to study the relationship between the Index of convexity of a compact set C in Rn and the Indices of its parallel bodies. Continuity properties of the Index are established relative to an appropriate metric on the class of compact sets in Rn.

Mathematical Subject Classification 2000
Primary: 52A20
Received: 7 May 1973
Revised: 25 November 1973
Published: 1 August 1974
Gerald A. Beer