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Polynomial bounds for Large Bernoulli sections of ℓ 1 N

AbstractWe present a quantitative form of the result of Bai and Yin from [2], and use to show that the section of ℓ1(1+δ)n spanned byn random independent sign vectors is with high probability...

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A Bernstein—Chernoff deviation inequality, and geometric properties of random...

AbstractIn this paper we first describe a new deviation inequality for sums of independent random variables which uses the precise constants appearing in the tails of their distributions, and can...

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Order Isomorphisms on Convex Functions in Windows

AbstractIn this paper we give a characterization of all order isomorphisms on some classes of convex functions. We deal with the class Cvx(K) consisting of lower-semi-continuous convex functions...

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On Multiplicative Maps of Continuous and Smooth Functions

AbstractIn 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...

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Order isomorphisms in cones and a characterization of duality for ellipsoids

AbstractWe study order isomorphisms in finite-dimensional ordered vector spaces. We generalize theorems of Alexandrov, Zeeman, and Rothaus (valid for “non-angular” cones) to wide classes of cones,...

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On weighted covering numbers and the Levi-Hadwiger conjecture

AbstractWe 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...

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On Godbersen’s conjecture

AbstractWe provide a natural generalization of a geometric conjecture of Fáry and Rédei regarding the volume of the convex hull of \(K \subset {\mathbb {R}}^n\), and its negative image \(-K\). We show...

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Differential analysis of polarity: Polar Hamilton-Jacobi, conservation laws,...

AbstractWe develop a differential theory for the polarity transform parallel to that of the Legendre transform, which is applicable when the functions studied are “geometric convex”, namely, convex,...

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Rogers–Shephard and local Loomis–Whitney type inequalities

AbstractWe provide functional analogues of the classical geometric inequality of Rogers and Shephard on products of volumes of sections and projections. As a consequence we recover (and obtain some...

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Functional Covering Numbers

AbstractWe define covering and separation numbers for functions. We investigate their properties, and show that for some classes of functions there is exact equality between separation and covering...

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