On the Reverse Loomis–Whitney Inequality

On the Reverse Loomis–Whitney Inequality The present paper deals with the problem of computing (or at least estimating) the $$\mathrm {LW}$$ LW -number $$\lambda (n)$$ λ ( n ) , i.e., the supremum of all $$\gamma $$ γ such that for each convex body K in $${\mathbb {R}}^n$$ R n there exists an orthonormal basis $$\{u_1,\ldots ,u_n\}$$ { u 1 , … , u n } such that $$\begin{aligned} {\text {vol}}_n(K)^{n-1} \ge \gamma \prod _{i=1}^n {\text {vol}}_{n-1} (K|u_i^{\perp }) , \end{aligned}$$ vol n ( K ) n - 1 ≥ γ ∏ i = 1 n vol n - 1 ( K | u i ⊥ ) , where $$K|u_i^{\perp }$$ K | u i ⊥ denotes the orthogonal projection of K onto the hyperplane $$u_i^{\perp }$$ u i ⊥ perpendicular to $$u_i$$ 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 $$\lambda (n)$$ λ ( n ) and deal with the problem of actually computing the $$\mathrm {LW}$$ LW -constant of a rational polytope. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Discrete & Computational Geometry Springer Journals

On the Reverse Loomis–Whitney Inequality

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Publisher
Springer US
Copyright
Copyright © 2017 by Springer Science+Business Media, LLC
Subject
Mathematics; Combinatorics; Computational Mathematics and Numerical Analysis
ISSN
0179-5376
eISSN
1432-0444
D.O.I.
10.1007/s00454-017-9949-9
Publisher site
See Article on Publisher Site

Abstract

The present paper deals with the problem of computing (or at least estimating) the $$\mathrm {LW}$$ LW -number $$\lambda (n)$$ λ ( n ) , i.e., the supremum of all $$\gamma $$ γ such that for each convex body K in $${\mathbb {R}}^n$$ R n there exists an orthonormal basis $$\{u_1,\ldots ,u_n\}$$ { u 1 , … , u n } such that $$\begin{aligned} {\text {vol}}_n(K)^{n-1} \ge \gamma \prod _{i=1}^n {\text {vol}}_{n-1} (K|u_i^{\perp }) , \end{aligned}$$ vol n ( K ) n - 1 ≥ γ ∏ i = 1 n vol n - 1 ( K | u i ⊥ ) , where $$K|u_i^{\perp }$$ K | u i ⊥ denotes the orthogonal projection of K onto the hyperplane $$u_i^{\perp }$$ u i ⊥ perpendicular to $$u_i$$ 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 $$\lambda (n)$$ λ ( n ) and deal with the problem of actually computing the $$\mathrm {LW}$$ LW -constant of a rational polytope.

Journal

Discrete & Computational Geometry – Springer Journals

Published: Oct 27, 2017

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