arXiv subjects

Galyna V. Livshyts

Publications and source records attributed to Galyna V. Livshyts.

At least 19 recordsLinked to original sources

arXiv2026

Blaschke--Santaló inequalities with respect to costs

We study functional Blaschke--Santaló inequalities in which the inner product is replaced by a cost. An inequality on each orthant, combined with Steiner symmetrization, gives a sufficient condition for a cost inequality to hold for even convex functions. We apply this principle to a multiplicative cost defined on all orthants. Its second variation gives a weighted improvement of the Brascamp--Lieb inequality in the directions for which differentiation is justified. We also prove an $L^p$ version of the Fradelizi--Meyer theorem from a weighted volume-product inequality and derive the corresponding radial cost and Brascamp--Lieb inequalities. Explicit examples include anisotropic power and Cauchy densities, with dilation equality cases and a sharp obstruction below the coordinate exponent two. For the spherical logarithmic cost we compute an exact entropy--logdet deficit decomposition and the second-order deficit transverse to ellipsoids. We also explain the transport-entropy formulation and distinguish the restricted spectral question from the scalar $(B)$ property.

math.MG↗

arXiv2026

A universal bound in the dimensional Brunn-Minkowski inequality for log-concave measures

We show that for any log-concave measure $μ$ on $\mathbb{R}^n$, any pair of symmetric convex sets $K$ and $L$, and any $λ\in [0,1],$ $$μ((1-λ) K+λL)^{c_n}\geq (1-λ) μ(K)^{c_n}+λμ(L)^{c_n},$$ where $c_n\geq n^{-4-o(1)}.$ This constitutes progress towards the dimensional Brunn-Minkowski conjecture (see Gardner, Zvavitch \cite{GZ}, Colesanti, L, Marsiglietti \cite{CLM}). Moreover, our bound improves for various special classes of log-concave measures.

math.MG↗

arXiv2025

On the polar of Schneider's difference body

In 1970, Schneider introduced the $m$th-order extension of the difference body $DK$ of a convex body $K\subset\mathbb R^n$, the convex body $D^m(K)$ in $\mathbb R^{nm}$. He conjectured that its volume is minimized for ellipsoids when the volume of $K$ is fixed. In this work, we solve a dual version of this problem: we show that the volume of the polar body of $D^m(K)$ is maximized precisely by ellipsoids. For $m=1$ this recovers the symmetric case of the celebrated Blaschke-Santaló inequality. We also show that Schneider's conjecture cannot be tackled using standard symmetrization techniques, contrary to this new inequality. As an application for our results, we prove Schneider's conjecture asymptotically á la Bourgain-Milman. We also consider a functional version.

math.MG↗

arXiv2025

On the Smallest Singular Value of Log-Concave Random Matrices

Let $A$ be an $N\times n$ random matrix whose entries are coordinates of an isotropic log-concave random vector in $\mathbb{R}^{Nn}$. We prove sharp lower tail estimates for the smallest singular value of $A$ in the following cases: (1) when $N=n$ and $A$ is drawn from an unconditional distribution, with no independence assumption; (2) when the columns of $A$ are independent and $N\geq n$; (3) when $A$ is sufficiently tall, that is $N\geq (1+λ)n$ for any positive constant $λ$.

math.PR↗

arXiv2024

Kohler-Jobin meets Ehrhard: the sharp lower bound for the Gaussian principal frequency while the Gaussian torsional rigidity is fixed, via rearrangements

In this note, we provide an adaptation of the Kohler-Jobin rearrangement technique to the setting of the Gauss space. As a result, we prove the Gaussian analogue of the Kohler-Jobin's resolution of a conjecture of Pólya-Szegö: when the Gaussian torsional rigidity of a (convex) domain is fixed, the Gaussian principal frequency is minimized for the half-space. At the core of this rearrangement technique is the idea of considering a "modified" torsional rigidity, with respect to a given function, and rearranging its layers to half-spaces, in a particular way; the Rayleigh quotient decreases with this procedure. We emphasize that the analogy of the Gaussian case with the Lebesgue case is not to be expected here, as in addition to some soft symmetrization ideas, the argument relies on the properties of some special functions; the fact that this analogy does hold is somewhat of a miracle.

math.AP↗

arXiv2023

The lower bound for Koldobsky's slicing inequality via random rounding

We study the lower bound for Koldobsky's slicing inequality. We show that there exists a measure $μ$ and a symmetric convex body $K \subseteq \mathbb{R}^n$, such that for all $ξ\in S^{n-1}$ and all $t\in \mathbb{R},$ $$μ^+(K\cap(ξ^{\perp}+tξ))\leq \frac{c}{\sqrt{n}}μ(K)|K|^{-\frac{1}{n}}.$$ Our bound is optimal, up to the value of the universal constant. It improves slightly upon the results of the first named author and Koldobsky which included a doubly-logarithmic error. The proof is based on an efficient way of discretizing the unit sphere.

math.MG↗

arXiv2023

Stability and the equality case in the B-theorem

In this paper, we show the stability, and characterize the equality cases in the strong B-inequality of Cordero-Erasquin, Fradelizi and Maurey \cite{B-conj}. As an application, we establish uniqueness of Bobkov's maximal Gaussian measure position from \cite{Bobkov-Mpos}.

math.MG↗

arXiv2022

On a conjectural symmetric version of Ehrhard's inequality

We formulate a plausible conjecture for the optimal Ehrhard-type inequality for convex symmetric sets with respect to the Gaussian measure. Namely, letting $J_{k-1}(s)=\int^s_0 t^{k-1} e^{-\frac{t^2}{2}}dt$ and $c_{k-1}=J_{k-1}(+\infty)$, we conjecture that the function $F:[0,1]\rightarrow\mathbb{R},$ given by $$F(a)= \sum_{k=1}^n 1_{a\in E_k}\cdot(β_k J_{k-1}^{-1}(c_{k-1} a)+α_k)$$ (with an appropriate choice of a decomposition $[0,1]=\cup_{i} E_i$ and coefficients $α_i, β_i$) satisfies, for all symmetric convex sets $K$ and $L,$ and any $λ\in[0,1]$, $$ F\left(γ(λK+(1-λ)L)\right)\geq λF\left(γ(K)\right)+(1-λ) F\left(γ(L)\right). $$ We explain that this conjecture is "the most optimistic possible", and is equivalent to the fact that for any symmetric convex set $K,$ its \emph{Gaussian concavity power} $p^s(K,γ)$ is greater than or equal to $p_s(RB^k_2\times \mathbb{R}^{n-k},γ),$ for some $k\in \{1,...,n\}$. We call the sets $RB^k_2\times \mathbb{R}^{n-k}$ round $k$-cylinders; they also appear as the conjectured Gaussian isoperimetric minimizers for symmetric sets, see Heilman \cite{Heilman}. In this manuscript, we make progress towards this question, and prove certain inequality for which the round k-cylinders are the only equality cases. As an auxiliary result on the way to the equality case characterization, we characterize the equality cases in the "convex set version" of the Brascamp-Lieb inequality, and moreover, obtain a quantitative stability version in the case of the standard Gaussian measure; this may be of independent interest.

math.MG↗

arXiv2022

New bounds on the minimal dispersion

We provide a new construction for a set of boxes approximating axis-parallel boxes of fixed volume in $[0, 1]^d$. This improves upper bounds for the minimal dispersion of a point set in the unit cube and its inverse in both the periodic and non-periodic settings in certain regimes. Up to double logarithmic factor, our bounds are sharp. We also apply our construction to $k$-dispersion.

math.MG↗

arXiv2021

The smallest singular value of heavy-tailed not necessarily i.i.d. random matrices via random rounding

We are concerned with the small ball behavior of the smallest singular value of random matrices. Often, establishing such results involves, in some capacity, a discretization of the unit sphere. This requires bounds on the norm of the matrix, and the latter bounds require strong assumptions on the distribution of the entries, such as bounded fourth moments (for a weak estimate), sub-gaussian tails (for a strong estimate), and structural assumptions such as mean zero and variance one. Recently, Rebrova and Tikhomirov developed a discretization procedure which does not rely on strong tail assumptions for the entries. However, their argument still required the structural assumptions of mean zero, variance one i.i.d. entries. In this paper, we discuss an efficient discretization of the unit sphere, which works with exponentially high probability, does not require any such structural assumptions, and, furthermore, does not require independence of the rows of the matrix. We show the existence of nets near the sphere, which compare values of any (deterministic) random matrix on the sphere and on the net via the regularized the Hilbert-Schmidt norm, which we introduce. Such refinement is a form of averaging, and enjoys strong large deviation properties. As a consequence we show sharp small ball estimates for the smallest singular value of square random matrices under mild assumptions, and for the random matrices with arbitrary aspect ratio.

math.PR↗

arXiv2021

On the local version of the Log-Brunn-Minkowski conjecture and some new related geometric inequalities

We prove that for any semi-norm $\|\cdot\|$ on $\mathbb{R}^n,$ and any symmetric convex body $K$ in $\mathbb{R}^n,$ \begin{equation}\label{ineq-abs2} \int_{\partial K} \frac{\|n_x\|^2}{\langle x,n_x\rangle}\leq \frac{1}{|K|}\left(\int_{\partial K} \|n_x\| \right)^2, \end{equation} and characterize the equality cases of this new inequality. The above would also follow from the Log-Brunn-Minkowski conjecture, if the latter was proven, and we believe that it may be of independent interest. We, furthermore, obtain an improvement of this inequality in some cases, involving the Poincare constant of $K.$ The conjectured Log-Brunn-Minkowski inequality is a strengthening of the Brunn-Minkowski inequality in the partial case of symmetric convex bodies, equivalent to the validity of the following statement: for all symmetric convex smooth sets $K$ in $\mathbb{R}^n$ and all smooth even $f:\partial K\rightarrow \mathbb{R},$ \begin{equation}\label{ineq-abs} \int_{\partial K} H_x f^2-\langle \mbox{II}^{-1}\nabla_{\partial K} f, \nabla_{\partial K} f\rangle +\frac{f^2}{\langle x,n_x\rangle}\leq \frac{1}{|K|}\left(\int_{\partial K} f \right)^2. \end{equation} In this note, we verify the above with the particular choice of speed function $f(x)=|\langle v,n_x\rangle|$, for all symmetric convex bodies $K$, where $v\in\mathbb{R}^n$ is an arbitrary vector.

math.MG↗

arXiv2020

The smallest singular value of inhomogeneous square random matrices

We show that for an $n\times n$ random matrix $A$ with independent uniformly anti-concentrated entries, such that $\mathbb{E} ||A||^2_{HS}\leq K n^2$, the smallest singular value $σ_n(A)$ of $A$ satisfies $$ P\left( σ_n(A)\leq \frac{\varepsilon}{\sqrt{n}} \right) \leq C\varepsilon+2e^{-cn},\quad \varepsilon \ge 0. $$ This extends earlier results of Rudelson and Vershynin, and Rebrova and Tikhomirov by removing the assumption of mean zero and identical distribution of the entries across the matrix, as well as the recent result of Livshyts, where the matrix was required to have i.i.d. rows. Our model covers "inhomogeneus" matrices allowing different variances of the entries, as long as the sum of the second moments is of order $O(n^2)$. In the past advances, the assumption of i.i.d. rows was required due to lack of Littlewood--Offord--type inequalities for weighted sums of non-i.i.d. random variables. Here, we overcome this problem by introducing the Randomized Least Common Denominator (RLCD) which allows to study anti-concentration properties of weighted sums of independent but not identically distributed variables. We construct efficient nets on the sphere with lattice structure, and show that the lattice points typically have large RLCD. This allows us to derive strong anti-concentration properties for the distance between a fixed column of $A$ and the linear span of the remaining columns, and prove the main result.

math.PR↗

arXiv2020

On the $L_p$-Brunn-Minkowski and dimensional Brunn-Minkowski conjectures for log-concave measures

We study several of the recent conjectures in regards to the role of symmetry in the inequalities of Brunn-Minkowski type, such as the $L_p$-Brunn-Minkowski conjecture of Böröczky, Lutwak, Yang and Zhang, and the Dimensional Brunn-Minkowski conjecture of Gardner and Zvavitch, in a unified framework. We obtain several new results for these conjectures. We show that when $K\subset L,$ the multiplicative form of the $L_p$-Brunn-Minkowski conjecture holds for Lebesgue measure for $p\geq 1-Cn^{-0.75}$, which improves upon the estimate of Kolesnikov and Milman in the partial case when one body is contained in the other. We also show that the multiplicative version of the $L_p$-Brunn-Minkowski conjecture for the standard Gaussian measure holds in the case of sets containing sufficiently large ball (whose radius depends on $p$). In particular, the Gaussian Log-Brunn-Minkowski conjecture holds when $K$ and $L$ contain $\sqrt{0.5 (n+1)}B_2^n.$ We formulate an a-priori stronger conjecture for log-concave measures, extending both the $L_p$-Brunn-Minkowski conjecture and the Dimensional one, and verify it in the case when the sets are dilates and the measure is Gaussian. We also show that the Log-Brunn-Minkowski conjecture, if verified, would yield this more general family of inequalities. Our results build up on the methods developed by Kolesnikov and Milman as well as Colesanti, Livshyts, Marsiglietti. We furthermore verify that the local version of these conjectures implies the global version in the setting of general measures, and this step uses methods developed recently by Putterman.

math.AP↗

arXiv2019

Remarks on the Rényi Entropy of a sum of IID random variables

In this note we study a conjecture of Madiman and Wang which predicted that the generalized Gaussian distribution minimizes the Rényi entropy of the sum of independent random variables. Through a variational analysis, we show that the generalized Gaussian fails to be a minimizer for the problem.

cs.IT↗

arXiv2019

On the Gardner-Zvavitch conjecture: symmetry in the inequalities of Brunn-Minkowski type

In this paper, we study the conjecture of Gardner and Zvavitch from \cite{GZ}, which suggests that the standard Gaussian measure $γ$ enjoys $\frac{1}{n}$-concavity with respect to the Minkowski addition of \textbf{symmetric} convex sets. We prove this fact up to a factor of 2: that is, we show that for symmetric convex $K$ and $L,$ $$ γ(λK+(1-λ)L)^{\frac{1}{2n}}\geq λγ(K)^{\frac{1}{2n}}+(1-λ)γ(L)^{\frac{1}{2n}}. $$ Further, we show that under suitable dimension-free uniform bounds on the Hessian of the potential, the log-concavity of even measures can be strengthened to $p$-concavity, with $p>0,$ with respect to the addition of symmetric convex sets.

math.AP↗

arXiv2019

On the stability of Brunn-Minkowski type inequalities

Log-Brunn-Minkowski inequality was conjectured by Boröczky, Lutwak, Yang and Zhang \cite{BLYZ}, and it states that a certain strengthening of the classical Brunn-Minkowski inequality is admissible in the case of symmetric convex sets. It was recently shown by Nayar, Zvavitch, the second and the third authors \cite{LMNZ}, that Log-Brunn-Minkowski inequality implies a certain dimensional Brunn-Minkowski inequality for log-concave measures, which in the case of Gaussian measure was conjectured by Gardner and Zvavitch \cite{GZ}. In this note, we obtain stability results for both Log-Brunn-Minkowski and dimensional Brunn-Minkowski inequalities for rotation invariant log-conave measures near a ball. Remarkably, the assumption of symmetry is only necessary for Log-Brunn-Minkowski stability, which emphasizes an important difference between the two conjectured inequalities. Also, we determine the infinitesimal version of the log-Brunn-Minkowski inequality. As a consequence, we obtain a strong Poincaré-type inequality in the case of unconditional convex sets, as well as for symmetric convex sets on the plane. Additionally, we derive an infinitesimal equivalent version of the B-conjecture for an arbitrary measure.

math.MG↗

arXiv2019

Some remarks about the maximal perimeter of convex sets with respect to probability measures

In this note we study the maximal perimeter of a convex set in $\mathbb{R}^n$ with respect to various classes of measures. Firstly, we show that for a probability measure $μ$ on $ \mathbb{R}^n$, satisfying very mild assumptions, there exists a convex set of $μ$-perimeter at least $C\frac{\sqrt{n}}{\sqrt[4]{Var|X|} \sqrt{\mathbb{E}|X|}}.$ This implies, in particular, that for any isotropic log-concave measure $μ$ one may find a convex set of $μ$- perimeter of order $n^{\frac{1}{8}}$. Secondly, we derive a general upper bound of $Cn|| f||^{\frac{1}{n}}_{\infty}$ on the maximal perimeter of a convex set with respect to any log-concave measure with density $f$ in an appropriate position. Our lower bound is attained for a class of distributions including the standard normal distribution. Our upper bound is attained, say, for a uniform measure on the cube. In addition, for isotropic log-concave measures we prove an upper bound of order $n^2$ for the maximal $μ$-perimeter of a convex set.

math.MG↗

arXiv2018

A note on the quantitative local version of the log-Brunn-Minkowski inequality

We prove that the log-Brunn-Minkowski inequality \begin{equation*} |λK+_0 (1-λ)L|\geq |K|^λ|L|^{1-λ} \end{equation*} (where $|\cdot|$ is the Lebesgue measure and $+_0$ is the so-called log-addition) holds when $K\subset\mathbb{R}^n$ is a ball and $L$ is a symmetric convex body in a suitable $C^2$ neighborhood of $K$.

math.DG↗