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# Inequalities for the surface area of projections of convex bodies

Published:2017-03-27

• Apostolos Giannopoulos,
Department of Mathematics, University of Athens, Panepistimioupolis 157-84, Athens, Greece
• Alexander Koldobsky,
Department of Mathematics, University of Missouri, Columbia, MO, USA
• Petros Valettas,
Department of Mathematics, University of Missouri, Columbia, MO, USA
 Format: LaTeX MathJax PDF

## Abstract

We provide general inequalities that compare the surface area $S(K)$ of a convex body $K$ in ${\mathbb R}^n$ to the minimal, average or maximal surface area of its hyperplane or lower dimensional projections. We discuss the same questions for all the quermassintegrals of $K$. We examine separately the dependence of the constants on the dimension in the case where $K$ is in some of the classical positions or $K$ is a projection body. Our results are in the spirit of the hyperplane problem, with sections replaced by projections and volume by surface area.
 Keywords: surface area, convex body, projection
 MSC Classifications: 52A20 - Convex sets in $n$ dimensions (including convex hypersurfaces) [See also 53A07, 53C45] 46B05 - unknown classification 46B05

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