Vol. 61, No. 1, 1975

Download this article
Download this article. For screen
For printing
Recent Issues
Vol. 328: 1  2
Vol. 327: 1  2
Vol. 326: 1  2
Vol. 325: 1  2
Vol. 324: 1  2
Vol. 323: 1  2
Vol. 322: 1  2
Vol. 321: 1  2
Online Archive
The Journal
Editorial Board
Submission Guidelines
Submission Form
Policies for Authors
ISSN: 1945-5844 (e-only)
ISSN: 0030-8730 (print)
Special Issues
Author Index
To Appear
Other MSP Journals
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