arXiv subjects

Mark Meyer

Publications and source records attributed to Mark Meyer.

8 recordsLinked to original sources

arXiv2026

Norm rigidity and equality cases for the Dyn--Farkhi inequality

For a convex body $K\subset\mathbb{R}^2$ that is symmetric with respect to the origin, and for a nonempty set $S\subset\mathbb{R}^2$, we study the $K$-Hausdorff distance from convex hull, defined by \begin{align*} d^{(K)}(S):=\sup_{x\in \text{conv}(S)}\inf_{s\in S}\|x-s\|_K, \end{align*} where $\|\cdot \|_K$ is the norm whose closed unit ball is $K$. We consider the problem of characterizing the origin symmetric convex bodies $K$ for which \begin{align*} d^{(K)}(A+B)^2\leq d^{(K)}(A)^2+d^{(K)}(B)^2 \end{align*} holds for all nonempty compact $A,B\subset\mathbb{R}^2$. We solve this problem, proving that this property holds if and only if $K$ is an ellipse centered at $0$. We then characterize the conditions for equality for this bound when $K$ is an ellipse.

math.MG↗

arXiv2026

L_p-Rogers--Shephard type inequalities for L_p-zonoids and symmetric bodies

We study generalizations of the classical Rogers--Shephard inequalities in the framework of Firey $L_p$-summation. We first consider the class of asymmetric $L_p$-zonoids. In this setting, we show that proving a sharp $L_p$-Rogers--Shephard inequality for asymmetric $L_p$-zonoids in $\mathbb{R}^n$ is equivalent to proving a sharp inequality between the volumes of projections of $B_q^m\cap \mathbb{R}^m_+$ and $B_q^m$ onto an $n$-dimensional subspace $E$, where $q$ is the H\"older conjugate of $p$. We conjecture that the inequality is sharp when the subspace $E$ is a coordinate subspace. We fully establish this inequality along with equality conditions in the case $p =2$. For general $p$, we prove it in the case $n=m-1$, $n=1$, and discuss several particular cases, including an averaged version and a local version of the inequality. We then turn to the setting of convex bodies having a center of symmetry. Rogers and Shephard also proved a sharp version of their inequality for bodies in this class. We conjecture a similar bound for the $L_p$-summation, and we establish our conjecture for the particular case of asymmetric $L_1$-zonoids, which, in particular, proves our conjecture in the planar case.

math.MG↗

arXiv2026

Equality cases for the $L_p$-Rogers--Shephard inequality in the plane and for locally anti-blocking bodies in $\mathbb{R}^n$

The classical Rogers--Shephard inequalities were extended to the Firey $L_p$-summation by Bianchini and Colesanti in the plane and by Zvavitch and the second and fourth authors for locally anti-blocking convex bodies in $\mathbb{R}^n$, leaving open the equality cases. We characterize the equality cases of these inequalities: in both cases, for $p>1$, equality holds if and only if the convex body is a simplex with one vertex at the origin.

math.MG↗

arXiv2026

A sharp $p$-subadditive bound for the $l_p$ Hausdorff distance from convex hull

We study the $\ell_p$ Hausdorff distance from convex hull, which for a compact set $A\subset \mathbb{R}^n$ is defined by \begin{align*} d^{(\ell_p)}(A):=\sup_{x\in \text{conv}(A)}\inf_{a\in A}\|x-a\|_p. \end{align*} In the planar case $n=2$, we study the problem of finding the optimal constant $C_p$ such that \begin{align*} d^{(\ell_p)}(A+B)^p\leq C_p\left(d^{(\ell_p)}(A)^p+d^{(\ell_p)}(B)^p\right) \end{align*} for all nonempty compact $A,B\subset\mathbb{R}^2$. We resolve this question, proving that \begin{align*} C_p=\max\{1,2^{p-2}\}. \end{align*}

math.MG↗

arXiv2024

The Dyn-Farkhi conjecture and the convex hull of a sumset in two dimensions

For a compact set $A$ in $\mathbb{R}^n$ the Hausdorff distance from $A$ to $\text{conv}(A)$ is defined by \begin{equation*} d(A):=\sup_{a\in\text{conv}(A)}\inf_{x\in A}|x-a|, \end{equation*} where for $x=(x_1,\dots,x_n)\in\mathbb{R}^n$ we use the notation $|x|=\sqrt{x_1^2+\dots+x_n^2}$. It was conjectured in 2004 by Dyn and Farkhi that $d^2$ is subadditive on compact sets in $\mathbb{R}^n$. In 2018 this conjecture was proved false by Fradelizi et al. when $n\geq3$. The conjecture can also be verified when $n=1$. In this paper we prove the conjecture when $n=2$ and in doing so we prove an interesting representation of the sumset $\text{conv}(A)+\text{conv}(B)$ for full dimensional compact sets $A,B$ in $\mathbb{R}^2$.

math.MG↗

arXiv2024

Measuring the convexity of compact sumsets with the Schneider non-convexity index

In recent work, Franck Barthe and Mokshay Madiman introduced the concept of the Lyusternik region, denoted by $\Lambda_{n}(m)$, to better understand volumes of sumsets. They gave a characterization of $\Lambda_{n}(2)$ (the volumes of compact sets in $\mathbb{R}^n$ when at most $m=2$ sets are added together) and proved that Lebesgue measure satisfies a fractional superadditive property. We attempt to imitate the idea of the Lyusternik region by defining a region based on the Schneider non-convexity index function, which was originally defined by Rolf Schneider in 1975. We call this region the Schneider region, denoted by $S_{n}(m)$. In this paper, we will give an initial characterization of the region $S_{1}(2)$ and in doing so, we will prove that the Schneider non-convexity index of a sumset $c(A_1+A_2)$ has a best lower bound in terms of $c(A_1)$ and $c(A_2)$. We will pose some open questions about extending this lower bound to higher dimensions and large sums. We will also show that, analogous to Lebesgue measure, the Schneider non-convexity index has a fractional subadditive property. Regarding the Lyusternik region, we will show that when the number of sets being added is $m\geq3$, that the region $\Lambda_{n}(m)$ is not closed, proving a new qualitative property for the region.

math.MG↗

arXiv2023

Equality conditions for the fractional superadditive volume inequalities

While studying set function properties of Lebesgue measure, F. Barthe and M. Madiman proved that Lebesgue measure is fractionally superadditive on compact sets in $\mathbb{R}^n$. In doing this they proved a fractional generalization of the Brunn-Minkowski-Lyusternik (BML) inequality in dimension $n=1$. In this paper we will prove the equality conditions for the fractional superadditive volume inequalites for any dimension. The non-trivial equality conditions are as follows. In the one-dimensional case we will show that for a fractional partition $(\mathcal{G},\beta)$ and nonempty sets $A_1,\dots,A_m\subseteq\mathbb{R}$, equality holds iff for each $S\in\mathcal{G}$, the set $\sum_{i\in S}A_i$ is an interval. In the case of dimension $n\geq2$ we will show that equality can hold if and only if the set $\sum_{i=1}^{m}A_i$ has measure $0$.

math.MG↗

arXiv2020

Photon-Driven Neural Path Guiding

Although Monte Carlo path tracing is a simple and effective algorithm to synthesize photo-realistic images, it is often very slow to converge to noise-free results when involving complex global illumination. One of the most successful variance-reduction techniques is path guiding, which can learn better distributions for importance sampling to reduce pixel noise. However, previous methods require a large number of path samples to achieve reliable path guiding. We present a novel neural path guiding approach that can reconstruct high-quality sampling distributions for path guiding from a sparse set of samples, using an offline trained neural network. We leverage photons traced from light sources as the input for sampling density reconstruction, which is highly effective for challenging scenes with strong global illumination. To fully make use of our deep neural network, we partition the scene space into an adaptive hierarchical grid, in which we apply our network to reconstruct high-quality sampling distributions for any local region in the scene. This allows for highly efficient path guiding for any path bounce at any location in path tracing. We demonstrate that our photon-driven neural path guiding method can generalize well on diverse challenging testing scenes that are not seen in training. Our approach achieves significantly better rendering results of testing scenes than previous state-of-the-art path guiding methods.

cs.GR↗