Mahler's conjecture and wavelets
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 \[ \| \varphi \|_ \infty E| X|\geq {\textstyle {1\over 2}}. \]
scientific article; zbMATH DE number 764226
scientific article