An inequality in Lp

An isoperimetric inequality on the lp balls

The normalised volume measure on the lp unit ball (1 ≤ p ≤ 2) satisfies the following isoperimetric inequality: the boundary measure of a set of measure a is at least cnã log(1/ã), where ã = min(a, 1− a). Résumé Nous prouvons une inégalité isopérimétrique pour la mesure uniforme Vp,n sur la boule unité de l n p (1 ≤ p ≤ 2). Si Vp,n(A) = a, alors V + p,n(A) ≥ cn ã log 1/ã, où V + p,n est la mesu...

A NOTE ON AN LP-BRUNN-MINKOWSKI INEQUALITY FOR CONVEX MEASURES IN THE UNCONDITIONAL CASE By

We consider a different L-Minkowski combination of compact sets in R than the one introduced by Firey and we prove an L-BrunnMinkowski inequality, p ∈ [0, 1], for a general class of measures called convex measures that includes log-concave measures, under unconditional assumptions. As a consequence, we derive concavity properties of the function t 7→ μ(t 1 pA), p ∈ (0, 1], for unconditional con...

Explicit ABS Solution of a Class of Linear Inequality Systems and LP Problems

A MAXIMAL Lp-INEQUALITY FOR STATIONARY SEQUENCES AND ITS APPLICATIONS

The paper aims to establish a new sharp Burkholder-type maximal inequality in Lp for a class of stationary sequences that includes martingale sequences, mixingales and other dependent structures. The case when the variables are bounded is also addressed, leading to an exponential inequality for a maximum of partial sums. As an application we present an invariance principle for partial sums of c...

THE Lp VERSION OF NEWMAN’S INEQUALITY FOR LACUNARY POLYNOMIALS

The principal result of this paper is the establishment of the essentially sharp Markov-type inequality ‖xP (x)‖Lp[0,1] ≤ 1/p+ 12  n ∑ j=0 (λj + 1/p)  ‖P‖Lp[0,1] for every P ∈ span{xλ0 , xλ1 , . . . , xλn} with distinct real exponents λj greater than −1/p and for every p ∈ [1,∞]. A remarkable corollary of the above is the Nikolskii-type inequality ‖yP (y)‖L∞ [0,1] ≤ 13  n ∑ j=0 (λj + 1...