arXiv subjects

Mariano Merzbacher

Publications and source records attributed to Mariano Merzbacher.

5 recordsLinked to original sources

arXiv2026

An Improved Volume Ratio Bound via Isotropic Positions

We show that, for every pair of convex bodies $K,L\subset\mathbb R^n$, $$ \operatorname{vr}(K,L)\leq C\sqrt{n\log(n+1)}. $$ The main point is to place $K$ and $L^\circ$ in isotropic position. We then consider a random orthogonal image of $L$ and control the corresponding operator norm by combining the isotropic mean-gauge estimate of Bizeul and Klartag with Letwin's recent dimension-free bound for the third-moment parameter appearing in their estimate. Our result improves the bound $ \operatorname{vr}(K,L)\leq C\sqrt n \log(n+1)$ proved by Giannopoulos and Hartzoulaki, which had remained the best general estimate for nearly two and a half decades.

math.MG↗

arXiv2022

On the volume ratio of projections of convex bodies

We study the volume ratio between projections of two convex bodies. Given a high-dimensional convex body $K$ we show that there is another convex body $L$ such that the volume ratio between any two projections of fixed rank of the bodies $K$ and $L$ is large. Namely, we prove that for every $1\leq k\leq n$ and for each convex body $K\subset \mathbb{R}^n$ there is a centrally symmetric body $L \subset \mathbb{R}^n$ such that for any two projections $P, Q: \mathbb{R}^n \to \mathbb{R}^n$ of rank $k$ one has $$ \mbox{vr}(PK, QL) \geq c \, \min\left\{\frac{ k}{ \sqrt{n}} \, \sqrt{\frac{1}{\log \log \log(\frac{n\log(n)}{k})}}, \, \frac{\sqrt{k}}{\sqrt{\log(\frac{n\log(n)}{k})}}\right\}, $$ where $c>0$ is an absolute constant. This general lower bound is sharp (up to logarithmic factors) in the regime $k\geq n^{2/3}$.

math.MG↗

arXiv2021

Continuous quantitative Helly-type results

Brazitikos' results on quantititative Helly-type theorems (for the volume and for the diameter) rely on the work of Srivastava on sparsification of John's decompositions. We change this technique by a stronger recent result due to Friedland and Youssef. This, together with an appropriate selection in the accuracy of the approximation, allow us to obtain Helly-type versions which are sensitive to the number of convex sets involved.

math.MG↗

arXiv2020

Asymptotic estimates for the largest volume ratio of a convex body

The largest volume ratio of given convex body $K \subset \mathbb{R}^n$ is defined as $$\mbox{lvr}(K):= \sup_{L \subset \mathbb{R}^n} \mbox{vr}(K,L),$$ where the $\sup$ runs over all the convex bodies $L$. We prove the following sharp lower bound $$c \sqrt{n} \leq \mbox{lvr}(K),$$ for every body $K$ (where $c>0$ is an absolute constant). This result improves the former best known lower bound, of order $\sqrt{\frac{n}{\log \log(n)}}$. We also study the exact asymptotic behavior of the largest volume ratio for some natural classes. In particular, we show that $\mbox{lvr}(K)$ behaves as the square root of the dimension of the ambient space in the following cases: if $K$ is the unit ball of an unitary invariant norm in $\mathbb{R}^{d \times d}$ (e.g., the unit ball of the $p$-Schatten class $S_p^d$ for any $1 \leq p \leq \infty$), $K$ is the the unit ball of the full/symmetric tensor product of $\ell_p$-spaces endowed with the projective or injective norm or $K$ is unconditional.

math.MG↗

arXiv2019

The minimal volume of simplices containing a convex body

Let $K \subset \mathbb R^n$ be a convex body with barycenter at the origin. We show there is a simplex $S \subset K$ having also barycenter at the origin such that $\left(\frac{vol(S)}{vol(K)}\right)^{1/n} \geq \frac{c}{\sqrt{n}},$ where $c>0$ is an absolute constant. This is achieved using stochastic geometric techniques. Precisely, if $K$ is in isotropic position, we present a method to find centered simplices verifying the above bound that works with very high probability. As a consequence, we provide correct asymptotic estimates on an old problem in convex geometry. Namely, we show that the simplex $S_{min}(K)$ of minimal volume enclosing a given convex body $K \subset \mathbb R^n$, fulfills the following inequality $$\left(\frac{vol(S_{min}(K))}{vol(K)}\right)^{1/n} \leq d \sqrt{n},$$ for some absolute constant $d>0$. Up to the constant, the estimate cannot be lessened.

math.MG↗