Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2506.15370

On polynomial inequalities for cone-volumes of polytopes

Tom Baumbach·Martin Henk
View original source ↗Read source document ↗Export metadata ↓

Abstract

Motivated by the discrete logarithmic Minkowski problem we study for a given matrix $U\in\mathbb{R}^{n\times m}$ its cone-volume set $C_{\tt cv}(U)$ consisting of all the cone-volume vectors of polytopes $P(U,b)=\{ x\in\mathbb{R}^n : U^\intercal x\leq b\}$, $b\in\mathbb{R}^n_{\geq 0}$. We will show that $C_{\tt cv}(U)$ is a path-connected semialgebraic set which extends former results in the planar case or for particular polytopes. Moreover, we define a subspace concentration polytope $P_{\tt scc}(U)$ which represents geometrically the subspace concentration conditions for a finite discrete Borel measure on the sphere. This is up to a scaling the basis matroid polytope of $U$, and these two sets, $P_{\tt scc}(U)$ and $C_{\tt cv}(U)$, also offer a new geometric point of view to the discrete logarithmic Minkowski problem.

Explore related subjects

math.MG

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Tom Baumbach, Martin Henk. 2026-06-18. On polynomial inequalities for cone-volumes of polytopes. https://arxiv.org/abs/2506.15370

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

An approximate counterexample to the Barker--Larman problem in dimension $4$

The Barker--Larman problem asks if a convex body $K \subseteq \mathbb R^n$ containing the Euclidean ball $\mathbb B_n$, such that all the sections of $K$ by hyperplanes tangent to $\mathbb B_n$ have constant $(n-1)$-dimensional volume, must necessarily be a Euclidean ball. In this paper we show a result pointing to a negative answer in dimension $4$. Taking $λ_0 = 4 \sqrt{3}π$ and any $N \in \mathbb N$, we obtain the existence of a family of convex bodies $K_{λ,N}$ with $λ\in (λ_0-r_N, λ_0 + r_N)$, such that the sections of $K_{λ,N}$ by hyperplanes tangent to the Euclidean ball, have area within $c |λ- λ_0|^{N+1}$ of $λ$, while the difference between outradius and inradius of $K_{λ,N}$ is larger than $C |λ- λ_0|$. The bodies $K_{λ,N}$ are constructed via radial functions as \[ρ_{K_{λ,N}}(t) = \cos\left( \sum_{n=0}^N \frac{(λ-λ_0)^n}{n!} φ_n(t) \right)^{-1},\] where $t \in [0,2π), λ\in \mathbb R$ and $φ_n$ are trigonometric polynomials that can be computed explicitly. The convergence of the inner power series when $N \to \infty$ (which is left open) would imply a negative answer to the Barker--Larman problem in dimension $4$. As an example we obtain a convex body whose outradius and inradius differ by more than $0.176$, and the area of the sections oscillate by less than $3 \times 10^{-7}$.

math.MG↗
arXiv2026

On convex bodies with rotationally symmetric planar projections

Let $n\ge 3$ and let $K\subset\mathbb R^n$ be a convex body. For a two-dimensional linear subspace $P\subset\mathbb R^n$, let $K|P$ be the orthogonal projection of $K$ onto $P$. We prove that if for every two-dimensional subspace $P$, the planar convex body $K|P$ has $q$-fold rotational symmetry up to translation, then for $q\ge4$, this forces $K$ to be an Euclidean ball. The case $q=3$ is exceptional: non-spherical examples exist.

math.MG↗
arXiv2026

A Spectral Proof of Khachiyan's Ellipsoid Conjecture

For a convex body $K\subset\mathbb{R}^n$, let $w(K)$ denote the volume of its maximum-volume inscribed ellipsoid. We prove that every closed halfspace $H$ whose boundary passes through the center of the maximizing ellipsoid satisfies \[ w(K\cap H)\le\frac{\sqrt e}{2}\,w(K). \] The constant is optimal uniformly over all dimensions, as witnessed by a family of circular cones, thereby establishing Khachiyan's conjecture. The proof converts containment, maximality, and the central-cut condition into algebraic constraints on positive definite matrices. Two complementary spectral bounds from a diagonal model extend to arbitrary center displacements through a directional rank-one estimate for fractional trace powers. Concavity determines their joint optimum. We also derive finite-dimensional bounds and a necessary condition for near equality, with self-contained supporting proofs and an alternative resolvent argument. An AI language model discovered the proof in a human-directed research process. Lean 4 with mathlib verifies the main theorem, sharpness, and supporting results.

math.MG↗