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Shiri Artstein-Avidan

Publications and source records attributed to Shiri Artstein-Avidan.

33 records · Page 2Linked to original sources
arXiv2017

Functional Covering Numbers

We define covering and separation numbers for functions. We investigate their properties, and show that for some classes of functions there is exact equality of separation and covering. We provide analogues for various geometric inequalities on covering numbers, such as volume bounds, bounds connected with Hadwiger's conjecture, and inequalities about M-positions for geometric log-concave functions. In particular, we obtain strong versions of M-positions for geometric log-concave functions.

math.FA↗
arXiv2016

Isotropic Measures and Maximizing Ellipsoids: Between John and Loewner

We define a one parameter family of positions of a convex body which interpolates between the John position and the Loewner position: for $r>0$, we say that $K$ is in maximal intersection position of radius $r$ if $\textrm{Vol}_{n}(K\cap rB_{2}^{n})\geq \textrm{Vol}_{n}(K\cap rTB_{2}^{n})$ for all $T\in SL_{n}$. We show that under mild conditions on $K$, each such position induces a corresponding isotropic measure on the sphere, which is simply a normalized Lebesgue measure on $r^{-1}K\cap S^{n-1}$. In particular, for $r_{M}$ satisfying $r_{M}^{n}κ_{n}=\textrm{Vol}_{n}(K)$, the maximal intersection position of radius $r_{M}$ is an $M$-position, so we get an $M$-position with an associated isotropic measure. Lastly, we give an interpretation of John's theorem on contact points as a limit case of the measures induced from the maximal intersection positions.

math.MG↗
arXiv2016

The Fundamental Theorems of Affine and Projective Geometry Revisited

The fundamental theorem of affine geometry is a classical and useful result. For finite-dimensional real vector spaces, the theorem roughly states that a bijective self-mapping which maps lines to lines is affine. In this note we prove several generalizations of this result and of its classical projective counterpart. We show that under a significant geometric relaxation of the hypotheses, namely that only lines parallel to one of a fixed set of finitely many directions are mapped to lines, an injective mapping of the space must be of a very restricted polynomial form. We also prove that under mild additional conditions the mapping is forced to be affine-additive or affine-linear. For example, we show that five directions in three dimensional real space suffice to conclude affine-additivity. In the projective setting, we show that n+2 fixed projective points in real n-dimensional projective space , through which all projective lines that pass are mapped to projective lines, suffice to conclude projective-linearity.

math.GM↗
arXiv2013

On weighted covering numbers and the Levi-Hadwiger conjecture

We define new natural variants of the notions of weighted covering and separation numbers and discuss them in detail. We prove a strong duality relation between weighted covering and separation numbers and prove a few relations between the classical and weighted covering numbers, some of which hold true without convexity assumptions and for general metric spaces. As a consequence, together with some volume bounds that we discuss, we provide a bound for the famous Levi-Hadwiger problem concerning covering a convex body by homothetic slightly smaller copies of itself, in the case of centrally symmetric convex bodies, which is qualitatively the same as the best currently known bound. We also introduce the weighted notion of the Levi-Hadwiger covering problem, and settle the centrally-symmetric case, thus also confirm Naszódi's equivalent fractional illumination conjecture in the case of centrally symmetric convex bodies (including the characterization of the equality case, which was unknown so far).

math.MG↗
arXiv2013

Analysis of polarity

We develop a differential theory for the polarity transform parallel to that for the Legendre transform, which is applicable when the functions studied are "geometric convex", namely convex, non-negative and vanish at the origin. This analysis may be used to solve a family of first order equations reminiscent of Hamilton--Jacobi and conservation law equations, as well as some second order Monge-Ampere type equations. A special case of the latter, that we refer to as the homogeneous polar Monge--Ampere equation, gives rise to a canonical method of interpolating between convex functions.

math.AP↗
arXiv2013

From Symplectic Measurements to the Mahler Conjecture

In this note we link symplectic and convex geometry by relating two seemingly different open conjectures: a symplectic isoperimetric-type inequality for convex domains, and Mahler's conjecture on the volume product of centrally symmetric convex bodies. More precisely, we show that if for convex bodies of fixed volume in the classical phase space the Hofer-Zehnder capacity is maximized by the Euclidean ball, then a hypercube is a minimizer for the volume product among centrally symmetric convex bodies.

math.MG↗
arXiv2013

A note on Santaló inequality for the polarity transform and its reverse

We prove a Santaló and a reverse Santaló inequality for the polarity transform, which was recently re-discovered by Artstein-Avidan and Milman, in the class consisting of (even) log-concave functions attaining their maximal value 1 at the origin, also called geometric log-cancave functions. The bounds are sharp up to the optimal universal constants.

math.FA↗
arXiv2013

Remarks about Mixed Discriminants and Volumes

In this note we prove certain inequalities for mixed discriminants of positive semi-definite matrices, and mixed volumes of compact convex sets in n-dimensions. Moreover, we discuss how the latter are related to the monotonicity of an information functional on the class of convex bodies, which is a geometric analogue of the classical Fisher information.

math.MG↗
arXiv2012

Bounds for Minkowski Billiard Trajectories in Convex Bodies

In this paper we use the Ekeland-Hofer-Zehnder symplectic capacity to provide several bounds and inequalities for the length of the shortest periodic billiard trajectory in a smooth convex body in ${\mathbb R}^{n}$. Our results hold both for classical billiards, as well as for the more general case of Minkowski billiards.

math.SG↗
arXiv2011

On Multiplicative Maps of Continuous and Smooth Functions

In this note, we study the general form of a multiplicative bijection on several families of functions defined on manifolds, both real or complex valued. In the real case, we prove that it is essentially defined by a composition with a diffeomorphism of the underlying manifold (with a bit more freedom in families of continuous functions). Our results in the real case are mostly simple extensions of known theorems. We then show that in the complex case, the only additional freedom allowed is complex conjugation. Finally, we apply those results to characterize the Fourier transform between certain function spaces.

math.CA↗
arXiv2006

The M-ellipsoid, Symplectic Capacities and Volume

In this work we bring together tools and ideology from two different fields, Symplectic Geometry and Asymptotic Geometric Analysis, to arrive at some new results. Our main result is a dimension-independent bound for the symplectic capacity of a convex body by its volume radius.

math.SG↗
arXiv2006

On Symplectic Capacities and Volume Radius

In this work we discuss a conjecture of Viterbo relating the symplectic capacity of a convex body and its volume. The conjecture states that among all 2n-dimensional convex bodies with a given volume the euclidean ball has maximal symplectic capacity. We present a proof of this fact up to a logarithmic factor in the dimension, and many classes of bodies for which this holds up to a universal constant.

math.SG↗
arXiv2006

Polynomial bounds for large Bernoulli sections of $\ell_1^N$

We prove a quantitative version of the bound on the smallest singular value of a Bernoulli covariance matrix (due to Bai and Yin). Then we use this bound, together with several recent developments, to show that the distance from a random (1-delta) n - dimensional section of ell_1^n, realised as an image of a sign matrix, to an Euclidean ball is polynomial in 1/delta (and independent of n), with high probability.

math.FA↗