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arXiv · 2410.23720

Affine Quermassintegrals and Even Minkowski Valuations

Georg C. Hofstätter·Philipp Kniefacz·Franz E. Schuster
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Abstract

It is shown that each continuous even Minkowski valuation on convex bodies of degree $1 \leq i \leq n - 1$ intertwining rigid motions is obtained from convolution of the $i$th projection function with a unique spherical Crofton distribution. In case of a non-negative distribution, the polar volume of the associated Minkowski valuation gives rise to an isoperimetric inequality which strengthens the classical relation between the $i$th quermassintegral and the volume. This large family of inequalities unifies earlier results obtained for $i = 1$ and $n - 1$. In these cases, isoperimetric inequalities for affine quermassintegrals, specifically the Blaschke-Santaló inequality for $i = 1$ and the Petty projection inequality for $i = n - 1$, were proven to be the strongest inequalities. An analogous result for the intermediate degrees is established here. Finally, a new sufficient condition for the existence of maximizers for the polar volume of Minkowski valuations intertwining rigid motions reveals unexpected examples of volume inequalities having asymmetric extremizers.

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Georg C. Hofstätter, Philipp Kniefacz, Franz E. Schuster. 2024-10-31. Affine Quermassintegrals and Even Minkowski Valuations. https://arxiv.org/abs/2410.23720

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