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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
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Abstract

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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

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[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