Hostname: page-component-77c89778f8-9q27g Total loading time: 0 Render date: 2024-07-17T04:58:07.749Z Has data issue: false hasContentIssue false
  • English
  • Français

On the measure of the one-skeleton of the sum of convex compact sets

Part of: General convexity

Published online by Cambridge University Press:  09 April 2009

Leoni Dalla
Leoni Dalla
Affiliation:
Department of MathematicsUniversity of AthensPanepistemiopolis 157 81 Athens, Greece
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

For any two compact convex sets in a Euclidean space, the relation between the volume of the sum of the two sets and the volume of each of them is given by the Brünn-Minkowski inequality. In this note we prove an analogous relation for the one-dimensional Hausdorff measure of the one-skeleton of the above sets. Also, some counterexamples are given which show that the above results are the best possible in some special cases.

MSC classification

Secondary: 52A20: Convex sets in $n$ dimensions (including convex hypersurfaces)
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 42 , Issue 3 , June 1987 , pp. 385 - 389
Copyright
Copyright © Australian Mathematical Society 1987

References

[1]Burton, G. R., The measure of the s–skeleton of a convex body, Mathematica 26 (1979), 290–301.Google Scholar
[2]Choquet, G., Lectures in Analysis, Vol. II (W. A. Benjamin, New York, Amsterdam, 1969).Google Scholar
[3]Eggleston, H. G., Convexity (Cambridge Univ. Press, 1958).Google Scholar
[4]Larman, D. G. and Rogers, C. A., The finite dimensional skeleton of a compact convex set, Bull. London Math. Soc. 5 (1973), 145–153.Google Scholar