arXiv subjects

Oscar Ortega-Moreno

Publications and source records attributed to Oscar Ortega-Moreno.

12 recordsLinked to original sources

arXiv2026

Aomoto interpolation and Coxeter systems

In this paper, we construct a Lagrange-type basis for the Aomoto space $AO(\mathcal A)$, naturally indexed by the chambers of the hyperplane arrangement $\mathcal A$. The construction relies on a dimension theorem of Orlik and Terao and yields an interpolation formula for elements of $AO(\mathcal A)$. We use this formula to characterize the extremal configurations in the strong polarization inequality as those arising from finite Coxeter reflection systems. We further show that the interpolation formula gives rise to a family of \emph{chamber identities}, including identities that were central to our earlier proof of the strong polarization problem and the Gaussian product inequality. Finally, we adapt the recent breakthrough of Ouimet and Greaves to prove a generalized Gaussian Product Inequality for completely monotone functions.

math.MG↗

arXiv2026

Polarization problems and Coxeter systems

In this paper we provide a proof of the strong polarization conjecture due to Ball and Frenkel. We observe that Coxeter systems produce unexpected new examples of extremizers. Furthermore, as a by-product of our work we solve the real polarization problem and completely characterize its extremal cases.

math.FA↗

arXiv2026

Strong Polarization and Entropy

We show that for any set of $n$ unit vectors $v_1,\ldots,v_n$ in a real Hilbert space and positive numbers $p_1,\ldots,p_n$ satisfying $\sum_j p_j = 1$, there exists a unit vector $u$ such that \[ \sum_{j=1}^n \frac{p_j^2}{\langle v_j, u\rangle^2}\leq 1. \] This inequality is a weighted version of the strong polarization inequality. As immediate corollaries, it yields a polarization inequality for products of powers of linear functionals and a strengthening of Bang's classical plank theorem for Hilbert spaces. The proof follows the approach introduced by Martínez and Ortega-Moreno in their recent solution to the strong polarization conjecture posed by Ball and Frenkel. We further note that our weighted inequality admits a Shannon-entropy interpretation: in a random sensing model, the entropy of the weights controls the minimum expected logarithmic loss.

math.FA↗

arXiv2026

The Christoffel problem for the disk area measure

The mixed Christoffel problem asks for necessary and sufficient conditions for a Borel measure on the Euclidean unit sphere to be the mixed area measure of some convex bodies, all but one of them are fixed. We consider the case in which the reference bodies are $(n-1)$-dimensional disks lying in a fixed hyperplane. We obtain an integral representation that reconstructs the support function of a convex body from its disk area measure, without any regularity assumptions. In the smooth setting, we reformulate the problem as a linear differential equation on the sphere, and derive a necessary and sufficient condition on the density of the disk area measure guaranteeing both convexity and regularity of the solution.

math.MG↗

arXiv2025

Mixed Christoffel-Minkowski problems for bodies of revolution

The mixed Christoffel-Minkowski problem asks for necessary and sufficient conditions for a Borel measure on the Euclidean unit sphere to be the mixed area measure of some convex bodies, one of which, appearing multiple times, is free and the rest are fixed. In the case where all bodies involved are symmetric around a common axis, we provide a complete solution to this problem, without assuming any regularity. In particular, we refine Firey's classification of area measures of figures of revolution. In our argument, we introduce an easy way to transform mixed area measures and mixed volumes involving axially symmetric bodies, and we significantly improve Firey's estimate on the local behavior of area measures. As a secondary result, we obtain a family of Hadwiger type theorems for convex valuations that are invariant under rotations around an axis.

math.MG↗

arXiv2024

The Klain approach to zonal valuations

We show an analogue of the Klain-Schneider theorem for valuations that are invariant under rotations around a fixed axis, called zonal. Using this, we establish a new integral representation of zonal valuations involving mixed area measures with a disk. In our argument, we introduce an easy way to translate between this representation and the one involving area measures, yielding a shorter proof of a recent characterization by Knoerr. As applications, we obtain various zonal integral geometric formulas, extending results by Hug, Mussnig, and Ulivelli. Finally, we provide a simpler proof of the integral representation of the mean section operators by Goodey and Weil.

math.MG↗

arXiv2024

Lefschetz operators on convex valuations

We investigate the action of Alesker's Lefschetz operators on translation invariant valuations on convex bodies. For scalar valued valuations, we describe this action on the level of Klain-Schneider functions by a Radon type transform, generalizing a result by Schuster and Wannerer. In the case of rotationally equivariant Minkowski valuations, the Lefschetz operators act on the generating function as a convolution transform. We show that the convolution kernel satisfies a Legendre type differential equation, and thus, is a strictly positive function that is smooth up to one point.

math.MG↗

arXiv2023

Fixed Points of Mean Section Operators

We characterize rotation equivariant bounded linear operators from $C(\mathbb{S}^{n-1})$ to $C^2(\mathbb{S}^{n-1})$ by the mass distribution of the spherical Laplacian of their kernel function on small polar caps. Using this characterization, we show that every continuous, homogeneous, translation invariant, and rotation equivariant Minkowski valuation $Φ$ that is weakly monotone maps the space of convex bodies with a $C^2$ support function into itself. As an application, we prove that if $Φ$ is in addition even or a mean section operator, then Euclidean balls are its only fixed points in some $C^2$ neighborhood of the unit ball. Our approach unifies and extends previous results by Ivaki from 2017 and the second author together with Schuster from 2021.

math.MG↗

arXiv2021

Iterations of Minkowski Valuations

It is shown that for any sufficiently regular even Minkowski valuation $Φ$ which is homogeneous and intertwines rigid motions, and for any convex body $K$ in a smooth neighborhood of the unit ball, there exists a sequence of positive numbers $(γ_m)_{m=1}^\infty$ such that $(γ_mΦ^m K)_{m=1}^\infty$ converges to the unit ball with respect to the Hausdorff metric.

math.MG↗

arXiv2021

The complex plank problem, revisited

Ball's complex plank theorem states that if $v_1,\dots,v_n$ are unit vectors in $\mathbb{C}^d$, and $t_1,\dots,t_n$, non-negative numbers satisfying $\sum_{k=1}^nt_k^2 = 1,$ then there exists a unit vector $v$ in $\mathbb{C}^d$ for which $|\langle v_k,v \rangle | \geq t_k$ for every $k$. Here we present a streamlined version of Ball's original proof.

math.FA↗

arXiv2021

Fixed points of Minkowski valuations

It is shown that for any sufficiently regular even Minkowski valuation $Φ$ which is homogeneous and intertwines rigid motions, there exists a neighborhood of the unit ball, where balls are the only solutions to the fixed-point problem $Φ^2 K = αK$. This significantly generalizes results by Ivaki for projection bodies and suggests, via the Lutwak--Schneider class reduction technique, a new approach to Petty's conjectured projection inequality.

math.MG↗

arXiv2020

An Optimal Plank Theorem

We give a new proof of Fejes Tóth's zone conjecture: for any sequence $v_1,v_2,...,v_n$ of unit vectors in a real Hilbert space $\mathcal{H}$, there exists a unit vector $v$ in $\mathcal{H}$ such that \begin{equation*} |\langle v_k,v \rangle| \geq \sin(π/2n) \end{equation*} for all $k$. This can be seen as sharp version of the plank theorem for real Hilbert spaces. Our approach is inspired by Ball's solution to the complex plank problem and thus unifies both the complex and the real solution under the same method.

math.FA↗