Mahler's conjecture and wavelets
type: Item
instance of: scholarly article
title: Mahler's conjecture and wavelets
zbMATH Open document ID: 0824.52005
DOI: 10.1007/BF02574044
author: Keith Ball
published in: Discrete \& Computational Geometry
publication date: 1995-07-02T00:00:00Z
full work available at URL: https://eudml.org/doc/131362
review text: Let \(k\) be a symmetric convex body in \(\mathbb{R}^ n\), and let \(K^*\) be its polar body. Mahler's conjecture asserts that \[ \text{vol } (K)\cdot \text{vol } (K^*) \geq {{4n} \over {n!}} \] where equality is attained for several pairs of bodies. The author considers Mahler's conjecture when \(K\) has \(2n+2\) facets. Such a \(K\) can be realized, up to affine invariance, as a one-codimensional section (through the centre) of an \((n+1)\)-dimensional cube. After discussing several interesting reformulations of Mahler's conjecture for that particular case, the author reduces this conjecture to another one concerning the scaling functions \(\varphi\) which appear in the construction of wavelets: Suppose \(P\) is the law of a symmetric random variable \(X\) and that there is a nonnegative integrable \(\varphi\) satisfying i) \(\int \varphi=1\) and ii) \(\varphi (y)= 2\int_{-\infty} ^{\infty} \varphi (2y-x) dP(x)\), \(y\in \mathbb{R}\) then \[ \| \varphi \|_ \infty E| X|\geq {\textstyle {1\over 2}}. \]
Mathematics Subject Classification ID: 52A20
Mathematics Subject Classification ID: 52A38
Mathematics Subject Classification ID: 52A40
zbMATH DE Number: 764226
zbMATH Keywords: polar bodies
zbMATH Keywords: cube slicing
zbMATH Keywords: Mahler's conjecture
zbMATH Keywords: wavelets
MaRDI profile type: Publication
Wikidata QID: Q122961150
cites work: Cube Slicing in R n
cites work: New volume ratio properties for convex symmetric bodies in \({\mathbb{R}}^ n\)
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cites work: Ten Lectures on Wavelets
cites work: Slicing the Cube in R n and Probability (Bounds for the Measure of a Central Cube Slice in R n by Probability Methods)
cites work: On the best constants in the Khinchin inequality
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description: scientific article; zbMATH DE number 764226