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.
