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On the Reverse Loomis–Whitney Inequality

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Abstract

The present paper deals with the problem of computing (or at least estimating) the \(\mathrm {LW}\)-number \(\lambda (n)\), i.e., the supremum of all \(\gamma \) such that for each convex body K in \({\mathbb {R}}^n\) there exists an orthonormal basis \(\{u_1,\ldots ,u_n\}\) such that

$$\begin{aligned} {\text {vol}}_n(K)^{n-1} \ge \gamma \prod _{i=1}^n {\text {vol}}_{n-1} (K|u_i^{\perp }) , \end{aligned}$$

where \(K|u_i^{\perp }\) denotes the orthogonal projection of K onto the hyperplane \(u_i^{\perp }\) perpendicular to \(u_i\). Any such inequality can be regarded as a reverse to the well-known classical Loomis–Whitney inequality. We present various results on such reverse Loomis–Whitney inequalities. In particular, we prove some structural results, give bounds on \(\lambda (n)\) and deal with the problem of actually computing the \(\mathrm {LW}\)-constant of a rational polytope.

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Acknowledgements

The authors are grateful to the referees for their valuable comments on a previous version of the present paper.

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Authors and Affiliations

  1. Dipartimento di Ingegneria dell’Informazione, Università degli Studi di Siena, 53100, Siena, Italy

    Stefano Campi

  2. Zentrum Mathematik, Technische Universität München, 85747, Garching bei München, Germany

    Peter Gritzmann

  3. Dipartimento di Matematica e Informatica “Ulisse Dini”, Università degli Studi di Firenze, 50122, Florence, Italy

    Paolo Gronchi

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  1. Stefano Campi
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Correspondence to Peter Gritzmann.

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Campi, S., Gritzmann, P. & Gronchi, P. On the Reverse Loomis–Whitney Inequality. Discrete Comput Geom 60, 115–144 (2018). https://doi.org/10.1007/s00454-017-9949-9

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