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On Benson’s Definition of Area in Minkowski Space

Published online by Cambridge University Press:  20 November 2018

A. C. Thompson
A. C. Thompson*
Affiliation:
Department of Mathematics, Statistics and Computing Science Dalhousie University Halifax, Nova Scotia B3H 3J5
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Abstract

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Let $(X,\,\left\| \,.\, \right\|)$ be a Minkowski space (finite dimensional Banach space) with unit ball $B$. Various definitions of surface area are possible in $X$. Here we explore the one given by Benson [1], [2]. In particular, we show that this definition is convex and give details about the nature of the solution to the isoperimetric problem.

Keywords

52A2152A38
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 42 , Issue 2 , 01 June 1999 , pp. 237 - 247
Copyright
Copyright © Canadian Mathematical Society 1999

References

[1] Benson, R. V., The Geometry of Affine Areas. Ph.D. Thesis, University of Southern California, Los Angeles, 1962. (University Microfilms Inc., Ann Arbor, Michigan 62-6037.)Google Scholar
[2] Benson, R. V., Euclidean Geometry and Convexity. McGraw-Hill, New York, 1966.Google Scholar
[3] Busemann, H., The isoperimetric problem for Minkowski area. Amer. J. Math. 71 (1949), 743–762.Google Scholar
[4] Busemann, H., The foundations of Minkowski geometry. Comment.Math. Helv. 24 (1950), 156–187.Google Scholar
[5] Busemann, H. and Straus, E. G., Area and normality. Pacific J. Math. 10 (1960), 35–72.Google Scholar
[6] Busemann, H., Ewald, G. and Shephard, G. C., Convex bodies and convexity on Grassmann cones IV. Math. Ann. 151 (1963), 35–41.Google Scholar
[7] Coxeter, H. S. M., Regular Polytopes. 3rd edn, Dover, New York, 1973.Google Scholar
[8] Danzer, L., Laugwitz, D. and Lenz, H., Ü ber das Löwnersche Ellipsoid und sein Analogon unter den einem Eikörper einbeschriebenen Ellipsoiden. Arch. Math. 8 (1957), 214–219.Google Scholar
[9] Gardner, R. J., Geometric Tomography. Encyclopedia Math. Appl. 58, Cambridge, New York, 1995.Google Scholar
[10] Goodey, P. R. and Weil, W., Zonoids and generalizations. In: Handbook of Convex Geometry (Eds. P. M. Gruber and J. M.Wills), Vol. B, North-Holland, Amsterdam, 1993, 1297–1326.Google Scholar
[11] Gruber, P. M.,Minimal ellipsoids and their duals. Rend.Circ. Mat. Palermo (2) 37 (1988), 35–64.Google Scholar
[12] Schneider, R., Convex bodies: the Brunn-Minkowski theory. Encyclopedia Math. Appl. 44, Cambridge, New York, 1993.Google Scholar
[13] Taylor, A. E., A geometric theorem and its application to biorthogonal systems. Bull. Amer. Math. Soc. 53 (1947), 614–616.Google Scholar
[14] Thompson, A. C.,Minkowski Geometry. Encyclopedia Math. Appl. 63, Cambridge, New York, 1996.Google Scholar