Extensions of reverse volume difference inequalities
Using methods from Fourier analysis, the authors establish volume difference inequalities, i.e., upper and lower bounds for \(|K|^{\frac{n-k}{n}}-|L|^{\frac{n-k}{n}}\) of the form \(C^k\min_{H\in G_{n,n-k}}(|K\cap H|-|L\cap H|)\) where \(G_{n,n-k}\) is the Grassmannian of \((n-k)\)-dimensional subspaces of \(\mathbb{R}^n\), \(|\cdot|\) is the volume, \(C\) is some constant, and \(K\), \(L\) are convex bodies satisfying a set of extra conditions. More precisely, the authors show that for \(1\leq k<n-3\) and \(K\subset\mathbb{R}^n\) an infinitely smooth origin-symmetric convex body with strictly positive curvature that is not a \(k\)-intersection body, there exists an origin-symmetric convex body \(L\subset K\) such that \[ |K|^{\frac{n-k}{n}}-|L|^{\frac{n-k}{n}}<C^k\min_{H\in G_{n,n-k}}(|K\cap H|-|L\cap H|) \] where the constant \(C\) depends on \(n\) and \(k\). Moreover, the authors prove that for \(1\leq k<n\) and \(L\subset\mathbb{R}^n\) an infinitely smooth origin-symmetric convex body with strictly positive curvature that is not a \(k\)-intersection body, there exists an origin-symmetric convex body \(K\supset L\) such that \[ |K|^{\frac{n-k}{n}}-|L|^{\frac{n-k}{n}}>C^k\min_{H\in G_{n,n-k}}(|K\cap H|-|L\cap H|) \] where the constant \(C\) depends on \(n\) and \(L\). For the entire collection see [Zbl 1392.52001].
scientific article; zbMATH DE number 6981359