Starshaped sets and the hausdorff metric

Abstract

Let C be a compact set in Rn. The ᵧ-parallel body of C, Bᵧ(C), is the union of the family of closed ᵧ-balls whose centers lie in C. If C is starshaped with respect to the origin, the gauge of Bᵧ(C) is a Lipschitz function; this observation in conjunction with the Arzela-Ascoli theorem yields Blaschke selection theorem for starshaped sets. In addition, each parallel body is a union of a finite collection of parallel bodies of starshaped sets. From this decomposition, we show that Lebesgue measure is continuous on the metric space of parallel bodies of a fixed radius in Rn relative to the Hausdorff metric. © 1975 Pacific Journal of Mathematics.