Vol. 61, No. 1, 1975

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ISSN: 0030-8730
Starshaped sets and the Hausdorff metric

Gerald A. Beer

Vol. 61 (1975), No. 1, 21–27

Let C be a compact set in Rn. The r-parallel body of C, Br(C), is the union of the family of closed r-balls whose centers lie in C. If C is starshaped with respect to the origin, the gauge of Br(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.

Mathematical Subject Classification 2000
Primary: 52A30
Received: 3 January 1975
Published: 1 November 1975
Gerald A. Beer