Further inequalities for the (generalized) Wills functional

The Wills functional \(\mathcal{W}(K)\) of a convex body \(K\), defined as the sum of its intrinsic volumes \(\mathrm{V}_i(K)\), turns out to have many interesting applications and properties. In this paper we make profit of the fact that it can be represented as the integral of a log-concave functi...

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Veröffentlicht in:arXiv.org 2020-02
Hauptverfasser: Alonso-Gutiérrez, David, Hernández Cifre, María A, Jesús Yepes Nicolás
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Sprache:eng
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Zusammenfassung:The Wills functional \(\mathcal{W}(K)\) of a convex body \(K\), defined as the sum of its intrinsic volumes \(\mathrm{V}_i(K)\), turns out to have many interesting applications and properties. In this paper we make profit of the fact that it can be represented as the integral of a log-concave function, which, furthermore, is the Asplund product of other two log-concave functions, and obtain new properties of the Wills functional (indeed, we will work in a more general setting). Among others, we get upper bounds for \(\mathcal{W}(K)\) in terms of the volume of \(K\), as well as Brunn-Minkowski and Rogers-Shephard type inequalities for this functional. We also show that the cube of edge-length 2 maximizes \(\mathcal{W}(K)\) among all \(0\)-symmetric convex bodies in John position, and we reprove the well-known McMullen inequality \(\mathcal{W}(K)\leq e^{\mathrm{V}_1(K)}\) using a different approach.
ISSN:2331-8422
DOI:10.48550/arxiv.1912.07993