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On maximal simplices inscribed in a central convex set

Part of: General convexity

Published online by Cambridge University Press:  26 February 2010

James R. McKinney
James R. McKinney
Affiliation:
California State Polytechnic University Pomona, California U.S.A.
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Extract

Blaschke [1] introduced the notion of maximal tetrahedra inscribed in two and three dimensional convex sets (maximal in the sense of volume). From this notion, he derived an inequality relating the volume of such maximal tetrahedra and the volume of the convex set, and used the inequality to characterize an ellipsoid and to obtain some results concerning isoperimetric inequalities.

MSC classification

Secondary: 52A20: Convex sets in $n$ dimensions (including convex hypersurfaces)
Type
Research Article
Information
Mathematika , Volume 21 , Issue 1 , June 1974 , pp. 38 - 44
Copyright
Copyright © University College London 1974

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References

1.Blaschke, W.. Vorlesungen über Differentialgeometrie I und II (Springer, Berlin, 1923).Google Scholar
2.Bonnesen, T. and Fenchel, W.. Theorie der konvexen Körper, Ergebnisse der mathematik, Vol. 3, No. 1 (Springer, Berlin, 1934).Google Scholar
3.Busemann, H.. The Geometry of Geodesies (Academic Press, New York, 1955).Google Scholar
4.Day, M. M.. “Polygons Circumscribed about Closed Convex Curves”, Trans. Amer. Math. Soc. 62 (1947), 315–319.CrossRefGoogle Scholar
5.Petty, C. M.. “On the Geometry of the Minkowski Plane”, Riv. Math. Univ. Parma, 6 (1955), 269–292.Google Scholar
6.Petty, C. M.. “Isoperimetric Problems”, Proc. Conference on Convexity and Combinatorial Geometry, Univ. of Oklahoma, June 1971, (1972), 26–41.Google Scholar
7.Taylor, A. E.. “A Geometric Theorem and its Applications to Biorthogonal Systems”, Bull. Amer. Math. Soc, 53 (1947), 614–616.Google Scholar