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 
The Alexanderov-Fenchel inequality asserts that
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
where 
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.
) by Cohn and Triantafillou, Li, and de Courcy-Ireland, Dostert, and Viazovska. I will also mention some other related papers. Two recent posts over here on sphere packing are 
of a binary code 
of length 
is defined as 
. Let us start with the following central questions:
, 
, what is (asymptotically) the maximum rate 
of a binary error-correcting binary code of length 
and a spherical 
-dimensional code 
. What is the maximum value of 
as 
for 
, what is (asymptotically) the best upper bound 
for 
that follows from Delsarte’s linear program.
of a binary error-correcting code of length 
in the 
let 
denote the entropy function. 
. In their paper 
goes to infinity, the LP bound on designs turns out, in conjunction with known lower bounds, to be proportional to 
. 
and is the current world record for 
.
(referred to as MRRW1) from the LP programs, went on to apply Delsarte’s linear program for constant weight codes and obtained similar upper bounds for them. Then they used these bounds to improve MRRW1 to stronger bounds (referred to as MRRW2) for some values of 
where 
is a vector chosen uniformly at random from the 5-dimensional simplex.
. A second way is to consider the parameters 
as being determined at random uniformly; and then taking the average over 
be an abstract graph with 
vertices. Lovasz and Yemini proved in 1982 that every 6-connected graph is 2-rigid, and made the conjecture that Villanyi has just proved. Jackson and Jordán proved in 2005 that 6-connected graphs are globally 2-rigid.
as 
This improves upon previous bounds for general 
and is the first asymptotically growing improvement to Rogers’ bound from 1947.
. (This is because when you double all spheres you will get a covering of space.) In 1905, this was improved by a factor of 
by Minkowski. The first asymptotic improvement was by Rogers in 1947 and he gave a lower bound of 
. Roger’s value for the constant 
was 
, and this value was improved to 
, 
, 
(when 
(when 
along a sparse sequence of dimensions. Here is the link to Venkatesh’s 
.
denote the maximum degree of a vertex in 
denote the maximum codegree, namely, the maximum number of common neighbours a pair of distinct vertices in G can have. (
is the independence number of 
and 
. Then
tends to 0 as 
and we can take 
and 
.
. The technique of proof is known as the Rodl nibble (or the semi-random method). Moving from Shearer-type theorems to results about sphere packing was pioneered twenty years ago by Krivelevich, Litsyn, and Vardy. (Their paper reproved a 
is a subset of 
has the set 
. The proof of the new sphere packing bound is based on discretization step, where you start with a huge ball 
and find a finite set of point 
such that 
. This is twice the value achieved in the paper, and the authors even predict that their graph theoretic theorem (and hence their sphere packing bound) can be improved by a factor 2. Roughly speaking, amorphous sphere packings have no long-term correlations.
of pairwise angle at least 
the maximal cardinality of such a spherical code. Let 
be the surface area of a spherical cup of radius 
(just look at a saturated family of spherical caps of radius 
) by a linear factor in the dimension. The new paper adds an additional 
lattice sphere packing is the densest sphere packing in eight dimensions.
, is the densest sphere packing in 
dimensions. Her proof relies on two inequalities between functions defined in terms of modular and quasimodular forms. We give a direct proof of these inequalities that does not rely on computer calculations.
one needs to find the following “magic function”:
is a real function on 
so that
depends only on 
, the distance from 
to the origin.
, where 
is the Fourier transform of 
for all 
for all 
.
. Here is what Dan told me about steps and ideas for Viazovska’s work. Before that Dan reminded me that a modular function of weight 
,
is a modular function of weight 4 (for Fourier eigenfunctions in dimension 8).
. At this point Dan shared his screen and started showing how to continue.
, and that all of these zeros should be of even order (as a function of ||x||), except for the zero at 
, which should be of odd order. The simplest way to satisfy these constraints is to look for a magic function with a zero of first order at 
and the simplest way to achieve that is to multiply by the factor 
. (This adds some complication, see below.)
also forces a double zero at 
, and Jacobi thetanull functions 
and 
is given below. Enjoy these beautiful formulas!
.
.
, 
of a real variable 
by
.
– numbers of polytopes and Hailun and her collaborators answer questions that I raised long ago whether the spece of stresses and the corresponding skeleton, determine the combinatorial structure, and at times, even the affine structure of a simplicial polytope. Patricia Hersh, described a relaxation of the notion of recursive atom ordering that still implies CL-shellability. These terms refer to famous notions of shellability of posets that were especially useful for the study of Coxeter groups. The work provides new avenues for proving shellability.
as the maximum number of lines all of their pairwise intersections are in 
. The conjecture is that the product of the measures is at least 
with illumination number at least 
, answering a question by G. Kalai. Furthermore, we prove the existence of finite sets of diameter 
which cannot be covered by 
balls of diameter 
so that every set of diameter 1 in 
.) There is also an 1982 elegant upper bound by Marek Lassak
. 
and both proofs give stronger results.
being replaced with 
.
.
balls of smaller diameter.) For covering by balls, it was known that the upper bound cannot be pushed down to 
balls of diameter 1 to cover an arbitrary set of diameter 1 in 
. The second theorem of the paper improves 1.0645 to 
which is 1.1547.
1.047… which stands for 
.
is said to be illuminated by a direction (unit vector) 
if the ray from 
The set of directions 
illuminate 
be the class of sets of diameter one of the form 
. It is interesting if by using such sets (of diameter 1) the upper bound of 
could be pushed down to 
, for some 
. It is also interesting if we can find a lower bound for covering with such sets of the form 
.
and Raigorodskii’s (world record) lower bound is 
. Raigorodskii’s (world record) bound for the double cup conjecture gives a bound of 
for the Borsuk problem. 
, namely, every set of diameter 1 can be covered by 
to 
).
Combinatorics and more 