Yair Shenfeld and Ramon van Handel Settled (for polytopes) the Equality Cases For The Alexandrov-Fenchel Inequalities

Combinatorics and more 2024-02-08

The power of negative thinking: Combinatorial and geometric inequalities

Two weeks ago, I participated (remotely) in the discrete geometry Oberwolfach meeting, and Ramon van Handel gave a beautiful lecture about the equality cases of Alexandrov-Fenchel inequalities which is among the most famous problems in convex geometry.

In the top left, Ramon explains the AF-inequality and its equality cases for three dimensions. The extremal bodies, also depicted in the top right, remind me of “stacked polytopes.” On the bottom right, you can see the AF-inequality, with the definition of mixed volumes above it. In the bottom left, there’s a discussion on equality cases and why they are challenging.

Following Ramon’s lecture, Igor Pak gave a presentation demonstrating that detecting cases of equality for the Alexandrov-Fenchel inequality is computationally hard. (If it is in the polynomial hierarchy then the hierarchy collapses.) Igor’s blog post (reblogged here) eloquently delves into this issue.

Here is some background. Given convex bodies $K_1,K_2,\dots,K_n$ in $\mathbb R^n$ , the volume of the Minkowski sum $t_1K_1+t_2K_2+\cdots +nK_n$ where $t_1,t_2,\dots,t_n\ge 0$ is a polynomial in $t_1,t_2,\dots, t_n$ . The mixed volume $V(K_1,K_2,\dots,K_n)$ is defined as the coefficient of the mixed term of this polynomial.

The Alexanderov-Fenchel inequality asserts that

$\displaystyle V(K_1,K_2,\dots,K_n)^2 \ge V(K_1,K_1,K_3,\dots,K_n)V(K_2,K_2,K_3,\dots ,K_n)$

And some overview: There exist intricate connections between three areas of convexity. The first concerns metric theory, particularly the study of valuations and the “Brunn-Minkowski” theory. The second involves arithmetic theory, specifically the “Eberhard theory” of counting integer points in dilations of integral polytopes. The third revolves around the combinatorial theory of faces, particularly the study of f-vectors. All these areas are related to algebraic geometry and commutative algebra, particularly to “Hodge theory.”

Here are links to the relevant papers:

Yair Shenfeld and Ramon van Handel, The Extremals of the Alexandrov-Fenchel Inequality for Convex Polytopes.

Abstract: The Alexandrov-Fenchel inequality, a far-reaching generalization of the classical isoperimetric inequality to arbitrary mixed volumes, lies at the heart of convex geometry. The characterization of its extremal bodies is a long-standing open problem that dates back to Alexandrov’s original 1937 paper. The known extremals already form a very rich family, and even the fundamental conjectures on their general structure, due to Schneider, are incomplete. In this paper, we completely settle the extremals of the Alexandrov-Fenchel inequality for convex polytopes. In particular, we show that the extremals arise from the combination of three distinct mechanisms: translation, support, and dimensionality. The characterization of these mechanisms requires the development of a diverse range of techniques that shed new light on the geometry of mixed volumes of nonsmooth convex bodies. Our main result extends further beyond polytopes in a number of ways, including to the setting of quermassintegrals of arbitrary convex bodies. As an application of our main result, we settle a question of Stanley on the extremal behavior of certain log-concave sequences that arise in the combinatorics of partially ordered sets.

Comments: I recall Rolf Schneider presenting (at Oberwolfach in the 1980s) his extensive class of examples for equality cases in A-F inequalities and formulating various conjectures about them. I also remember the relations discovered by Bernard Teissier and Askold Khovanskii between A-F inequalities and algebraic geometry. This connection is discussed in the last section of the paper.

Yair Shenfeld and Ramon van Handel, The Extremals of Minkowski’s Quadratic Inequality

Abstract: In a seminal paper “Volumen und Oberfläche” (1903), Minkowski introduced the basic notion of mixed volumes and the corresponding inequalities that lie at the heart of convex geometry. The fundamental importance of characterizing the extremals of these inequalities was already emphasized by Minkowski himself, but has to date only been resolved in special cases. In this paper, we completely settle the extremals of Minkowski’s quadratic inequality, confirming a conjecture of R. Schneider. Our proof is based on the representation of mixed volumes of arbitrary convex bodies as Dirichlet forms associated to certain highly degenerate elliptic operators. A key ingredient of the proof is a quantitative rigidity property associated to these operators.

Comment: For this special case of the A-F inequality Shenfeld and van Handel gave a complete characterization of the extremal cases. A special case of this special case asserts that

$S^2(K) \ge W(K)V(K)$

where $V(K),S(K),W(K)$ are the volume, surface area, and mean width of $K$ , respectively.

Swee Hong Chan and Igor Pak, Equality cases of the Alexandrov–Fenchel inequality are not in the polynomial hierarchy

Abstract: Describing the equality conditions of the Alexandrov–Fenchel inequality has been a major open problem for decades. We prove that in the case of convex polytopes, this description is not in the polynomial hierarchy unless the polynomial hierarchy collapses to a finite level. This is the first hardness result for the problem, and is a complexity counterpart of the recent result by Shenfeld and van Handel (arXiv:archive/201104059), which gave a geometric characterization of the equality conditions. The proof involves Stanley’s order polytopes and employs poset theoretic technology.

Igor Pak’s blog post elaborate on this paper. Three more (among quite a few) relevant papers:

Yair Shenfeld and Ramon van Handel, Mixed volumes and the Bochner method; D. Cordero-Erausquin, B. Klartag, Q. Merigot, F. Santambrogio, One more proof of the Alexandrov-Fenchel inequality ; Ramon van Handel, Shephard’s inequalities, Hodge-Riemann relations, and a conjecture of Fedotov.