On affine invariant and local Loomis-Whitney type inequalities

On a Loomis-Whitney Type Inequality for Permutationally Invariant Unconditional Convex Bodies

For a permutationally invariant unconditional convex body K in R^n we define a finite sequence (K_j), j = 1, ..., n of projections of the body K to the space spanned by first j vectors of the standard basis of R^n. We prove that the sequence of volumes (|K_1|, ..., |K_n|) is log-concave.

Reverse Loomis-Whitney inequalities via isotropicity

Given a centered convex body $K\subseteq\mathbb{R}^n$, we study the optimal value of the constant $\tildeΛ(K)$ such that there exists an orthonormal basis $\{w_i\}_{i=1}^n$ for which the following reverse dual Loomis-Whitney inequality holds: $$ |K|^{n-1}\leqslant \tildeΛ(K)\prod_{i=1}^n|K\cap w_i^\perp|. $$ We prove that $\tildeΛ(K)\leqslant(CL_K)^n$ for some absolute $C>1$ and that this estimate in terms of $L_K$, the isotropic constant of $K$, is asymptotically sharp in the sense that there exists another absolute constant $c>1$ and a convex body $K$ such that $(cL_K)^n\leqslant\tildeΛ(K)\l…

Rogers-Shephard and local Loomis-Whitney type inequalities

We 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 new) functional versions of Rogers-Shephard type inequalities as well as some generalizations of the geometric Rogers-Shephard inequality in the case where the subspaces intersect. These generalizations can be regarded as sharp local reverse Loomis-Whitney inequalities. We also obtain a sharp local Loomis-Whitney inequality.

On Whitney type inequalities for local anisotropic polynomial approximation

We prove a multivariate Whitney type theorem for the local anisotropic polynomial approximation in $L_p(Q)$ with $1\leq p\leq \infty$. Here $Q$ is a $d$-parallelepiped in $\RR^d$ with sides parallel to the coordinate axes. We consider the error of best approximation of a function $f$ by algebraic polynomials of fixed degree at most $r_i - 1$ in variable $x_i,\ i=1,...,d$, and relate it to a so-called total mixed modulus of smoothness appropriate to characterizing the convergence rate of the approximation error. This theorem is derived from a Johnen type theorem on equivalence between a certain…

Multilinear Duality and Factorisation for Brascamp-Lieb-type Inequalities with applications

We initiate the study of a duality theory which applies to norm inequalities for pointwise weighted geometric means of positive operators. The theory finds its expression in terms of certain pointwise factorisation properties of function spaces which are naturally associated to the norm inequality under consideration. We relate our theory to the Maurey-Nikisin-Stein theory of factorisation of operators, and present a fully multilinear version of Maurey's fundamental theorem on factorisation of operators through $L^1$. The development of the theory involves convex optimisation and minimax theor…

伴随 Brascamp-Lieb 不等式

Adjoint Brascamp-Lieb inequalities

本文提出 Brascamp-Lieb 不等式的伴随形式，将其刻画为熵版本的 $L^p$ 对应物，并用于重证 Gowers 范数对数凸性及建立层析变换的反向不等式。