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A characterization of all loglinear inequalities for three quermassintegrals of convex bodies


Author: Peter Gritzmann
Journal: Proc. Amer. Math. Soc. 104 (1988), 563-570
MSC: Primary 52A40
DOI: https://doi.org/10.1090/S0002-9939-1988-0962829-5
MathSciNet review: 962829
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Abstract: We give a complete characterization of all inequalities of the type $ W_i^\alpha (K)W_j^\beta (K)W_k^\gamma (K) \geq c$, where $ K$ is an arbitrary convex body of Euclidean $ d$-space, $ {W_l}(K),l = i,j,k$, denotes the $ l$th quermassintegral of $ K$ and $ \alpha ,\beta ,\gamma $ and $ c$ are arbitrary reals. A special case of such inequalities is the classical isoperimetric inequality for the volume and surface area of convex bodies. It turns out that all nontrivial inequalities of this type can be generated by means of $ d - 1$ basic inequalities, the Fenchel-Alexandrov inequalities.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1988-0962829-5
Keywords: Isoperimetric inequality, quermassintegrals, convex bodies, Fenchel-Alexandrov inequalities
Article copyright: © Copyright 1988 American Mathematical Society

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