Slicing \(\ell_p\)-balls reloaded: stability, planar sections in \(\ell_1\)
Let \(n\) be an integer; for a real number \(p>0\), let \[ B^n_p=\left\{x=(x_1,\dots,x_n)\in\mathbb{R}^n:\sum_{i=1}^n|x_i|^p\leq1\right\}. \] be the unit \(\ell_p\)-ball in the \(n\)-dimensional Euclidean space \(\mathbb{R}^n\). In particular, for \(p=1\), \(B^n_1\) is the so-called \textit{cross-polytope}. Let \(k\in\{1,\dots,n\}\) and \(H\) be an hyperplane of \(\mathbb{R}^n\) of dimension \(k\). The \(k\)-dimensional Hausdorff measure of the intersection of \(B^n_p\) with \(H\) is denoted by \(\operatorname{vol}_k(B^n_p\cap H)\). The problem of finding the maximum and minimum values of \(\operatorname{vol}_k(B^n_p\cap H)\), as \(H\) ranges over all possible \(k\)-dimensional hyperplanes, for various values of \(k\) and \(p\), is well studied. New results are provided by the authors in this paper. The first result is a sharp lower bound when \(p=1\) and \(k=2\). Theorem. Let \(n\geq 3\); for every \(2\)-dimensional hyperplane \(H\) \[ \operatorname{vol}_2(B^n_1\cap H)\geq \frac{n^2\sin^3\left(\frac{\pi}{2n}\right)}{\cos\left(\frac\pi{2n}\right)}. \] If equality holds in the previous inequality, then \(B_1^n\cap H\) is isometric to a regular planar \(2n\)-gon. The minimum is achieved for \(H=T(\mathbb{R}^2)\), where \(T:\mathbb{R}^2\to\mathbb{R}^n\) is defined by \(T(x)=(\langle x,v_1\rangle,\dots,\langle x,v_n\rangle)\), and \(v_k=\left(\cos\left(\frac{k\pi}{n}\right),\sin\left(\frac{k\pi}{n}\right)\right)\), for \(k=1,\dots,n\). The minimizing subspace is unique up to permutations of the coordinates, and reflections. The second main contribution of the paper is a stability result for the known cases of subspaces minimizing or maximizing the volume of their intersection with \(B^n_p\). Theorem. Let \(n\geq 1\). There exists a positive constant depending on \(p\), \(c_p\), such that for every unit vector \(a=(a_1,\dots,a_n)\), with \(a_1\geq a_2\geq \dots\geq a_n\), the following inequalities hold \begin{align*} \frac{\operatorname{vol}_{n-1}(B_p^n\cap a^\perp)}{\operatorname{vol}_{n-1}(B_p^n\cap e_1^\perp)} &\leq (a_1^p+(1-a_1^2)^{p/2})^{-1/p},\text{ if }0<p<2,\\ \frac{\operatorname{vol}_{n-1}(B_p^n\cap a^\perp)}{\operatorname{vol}_{n-1}\left(B_p^n\cap\left(\frac{e_1+\dots+e_n}{\sqrt n}\right)^\perp\right)} &\leq 1+c_p\sum_{j=1}^n(a_j^2-1/n)^2,\text{ if }0<p<2. \end{align*} Moreover \begin{align*} \frac{\operatorname{vol}_{n-1}(B_p^n\cap a^\perp)}{\operatorname{vol}_{n-1}(B_p^n\cap e_1^\perp)} &\geq 1+c_p|a-e_1|^2,\text{ if }2< p\le\infty,\\ \frac{\operatorname{vol}_{n-1}(B_\infty^n\cap a^\perp)}{\operatorname{vol}_{n-1}\left(B_\infty^n\cap\left(\frac{e_1+e_2}{\sqrt 2}\right)^\perp\right)} &\leq 1-c_\infty\left|a-\frac{e_1+e_2}{\sqrt2}\right|. \end{align*}
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