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TWO OPTIMISATION PROBLEMS FOR CONVEX BODIES

Part of: General convexity

Published online by Cambridge University Press:  05 August 2015

YUNLONG YANG  and
DEYAN ZHANG
YUNLONG YANG*
Affiliation:
Department of Mathematics, Tongji University, Shanghai 200092, PR China email 88ylyang@tongji.edu.cn
DEYAN ZHANG
Affiliation:
Department of Mathematics, Tongji University, Shanghai 200092, PR China email zhangdy8005@126.com
*
✉88ylyang@tongji.edu.cn
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Abstract

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In this paper, we will show that the spherical symmetric slices are the convex bodies that maximise the volume, the surface area and the integral of mean curvature when the minimum width and the circumradius are prescribed and the symmetric $2$-cap-bodies are the ones which minimise the volume, the surface area and the integral of mean curvature given the diameter and the inradius.

Keywords

spherical symmetric slicesymmetric 2-cap-bodyminimum widthdiameterinradiuscircumradius

MSC classification

Primary: 52A40: Inequalities and extremum problems
Secondary: 52A15: Convex sets in $3$ dimensions (including convex surfaces)
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 93 , Issue 1 , February 2016 , pp. 137 - 145
Copyright
© 2015 Australian Mathematical Publishing Association Inc. 

References

Blaschke, W., ‘Eine Frage über konvexe Körper’, Jahresber. Dtsch. Math.-Ver. 25 (1916), 121–125.Google Scholar
Bonnesen, T. and Fenchel, W., Theory of Convex Bodies (eds. Boron, L., Christenson, C. and Smith, B.) (BCS Associates, Moscow, Idaho, 1987).Google Scholar
Böröczky, K. Jr., Hernández Cifre, M. A. and Salinas, G., ‘Optimizing area and perimeter of convex sets for fixed circumradius and inradius’, Monatsh. Math. 138(2) (2003), 95–110.CrossRefGoogle Scholar
Burago, Yu. D. and Zelgaller, V. A., Geometric Inequalities (Springer, Berlin, 1988) (translated from the Russian by A. B. Sosinskiǐ).CrossRefGoogle Scholar
Finch, S. R., ‘Oblique circular cones and cylinders’, Preprint, 2012, arXiv:1212.5946, http://arxiv.org/pdf/1212.5946v2.pdf.Google Scholar
Hadwiger, H., Altes und Neues über konvexe Körper (Birkhäuser, Basel and Stuttgart, 1955) (in German).CrossRefGoogle Scholar
Henk, M. and Tsintsifas, G. A., ‘Some inequalities for planar convex figures’, Elem. Math. 49(3) (1994), 120–125.Google Scholar
Hernández Cifre, M. A., ‘Is there a planar convex set with given width, diameter, and inradius?’, Amer. Math. Monthly 107(10) (2000), 893–900.CrossRefGoogle Scholar
Hernández Cifre, M. A., ‘Optimizing the perimeter and the area of convex sets with fixed diameter and circumradius’, Arch. Math. (Basel) 79(2) (2002), 147–157.CrossRefGoogle Scholar
Hernández Cifre, M. A., Pastor, J. A., Salinas Martínez, G. and Segura Gomis, S., ‘Complete systems of inequalities for centrally symmetric convex sets in the n-dimensional space’, Arch. Inequal. Appl. 1(2) (2003), 155–167.Google Scholar
Hernández Cifre, M. A., Salinas, G. and Segura Gomis, S., ‘Complete systems of inequalities’, JIPAM. J. Inequal. Pure Appl. Math. 2(1) (2001), Article 10, 12 pages.Google Scholar
Hernández Cifre, M. A., Salinas, G. and Segura Gomis, S., ‘Two optimization problems for convex bodies in the n-dimensional space’, Beitr. Algebra Geom. 45(2) (2004), 549–555.Google Scholar
Hernández Cifre, M. A. and Segura Gomis, S., ‘The missing boundaries of the Santaló diagrams for the cases (d, w, R) and (w, R, r)’, Discrete Comput. Geom. 23(3) (2000), 381–388.CrossRefGoogle Scholar
Santaló, L. A., ‘On complete systems of inequalities between elements of a plane convex figure’, Math. Notae 17 (1961), 82–104 (in Spanish).Google Scholar
Schneider, R., Convex bodies: The Brunn–Minkowski Theory (Cambridge University Press, Cambridge, 1993).CrossRefGoogle Scholar
Scott, P. R., ‘A family of inequalities for convex sets’, Bull. Aust. Math. Soc. 20(2) (1979), 237–245.CrossRefGoogle Scholar
Scott, P. R. and Awyong, P. W., ‘Inequalities for convex sets’, JIPAM. J. Inequal. Pure Appl. Math. 1(1) (2000), Article 6, 6 pages.Google Scholar