Volumes of restricted Minkowski sums and the free analogue of the entropy power inequality

In noncommutative probability theory independence can be based on free products instead of tensor products. This yields a highly noncommutative theory: free probability theory (for an introduction see [9]). The analogue of entropy in the free context was introduced by the second named author in [8]. Here we show that Shannon's entropy power inequality ([6, 1]) has an analogue for the free entropy (X) (Theorem 2.1).The free entropy, consistent with Boltzmann's formulaS=klogW, was defined via volumes of matricial microstates. Proving the free entropy power inequality naturally becomes a geometric question.Restricting the Minkowski sum of two sets means to specify the set of pairs of points which will be added. The relevant inequality, which holds when the set of addable points is sufficiently large, differs from the Brunn-Minkowski inequality by having the exponent 1/n replaced by 2/n. Its proof uses the rearrangement inequality of Brascamp-Lieb-Lüttinger ([2]). Besides the free entropy power inequality, note that the inequality for restricted Minkowski sums may also underlie the classical Shannon entropy power inequality (see 3.2 below).Research supported in part by grants from the National Science Foundation.     

Saturating constructions for normed spaces

We prove several results of the following type: given finite dimensional normed space V there exists another space X with log dim X = O(log dim V) and such that every subspace (or quotient) of X, whose dimension is not too small, contains a further subspace isometric to V. This sheds new light on the structure of such large subspaces or quotients (resp. large sections or projections of convex bodies) and allows us to solve several problems stated in the 1980s by V. Milman.     

On the existence of an invariant measure for Markov-Feller operators

Let X be a Polish space and P a Markov operator acting on the space of Borel measures on X. We will prove the existence of an invariant measure with respect to P, provided that P satisfies some condition of a Prokhorov type and that the family of functions is equi-continuous with respect to the Prokhorov distance at some point of the space X. Moreover, we will construct a counterexample which show that the above equi-continuity condition cannot be dropped.     

Calderón–Zygmund Operators in the Bessel Setting for All Possible Type Indices

We show that many harmonic analysis operators in the Bessel setting,including maximal operators,Littlewood–Paley–Stein type square functions,multipliers of Laplace or Laplace–Stieltjes transform type and Riesz transforms are,or can be viewed as,Calderón–Zygmund operators for all possible values of type parameter λ in this context.This extends results existing in the literature,but being justified only for a restricted range of λ.     

Polarization justified Fukui functions: the theory and applications for molecules

The Fukui functions based on the computable local polarizability vector have been presented for a group of simple molecules. The necessary approximation for the density functional theory softness kernel has been supported by a theoretical analysis unifying and generalizing early concepts produced by the several authors. The exact relation between local polarizability vector and the derivative of the nonlocal part of the electronic potential over the electric field has been demonstrated. The resulting Fukui functions are unique and represent a reasonable refinement when compared to the classical ones that are calculated as the finite difference of the density in molecular ions. The new Fukui functions are strongly validated by their direct link to electron dipole polarizabilities that are reported experimentally and by other computational methods.     

More on the duality conjecture for entropy numbers

We verify, up to a logarithmic factor, the duality conjecture for entropy numbers in the case where one of the bodies is an ellipsoid. To cite this article: S. Artstein et al., C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Saturating constructions for normed spaces II

We prove several results of the following type: given finite-dimensional normed space V possessing certain geometric property there exists another space X having the same property and such that (1) and (2) every subspace of X, whose dimension is not “too small”, contains a further well-complemented subspace nearly isometric to V. This sheds new light on the structure of large subspaces or quotients of normed spaces (resp., large sections or linear images of convex bodies) and provides definitive solutions to several problems stated in the 1980s by Milman.