The role of inequalities in information theory is reviewed and the relationship of these inequalities to inequalities in other branches of mathematics is developed. Index Terms -Information inequalities, entropy power, Fisher information, uncertainty principles. I. PREFACE:~NEQUALITIES ININFORMATIONTHEORY I NEQUALITIES in information theory have been driven by a desire to solve communication th...

Then dp is a metric. To prove this one must check the axioms. First, since |xk − yk| = |yk − xk| ≥ 0, it is obvious that dp(x, y) = dp(y, x) ≥ 0 for all x and y. Furthermore, since ∑ k |xk−yk| = 0 if and only if all the terms |xk−yk| are zero, we see that dp(x, y) = 0 if and only if x = y. To verify the remaining axiom we use Minkowski’s Inequality. Let x, y, z ∈ C, and define ak = yk − xk and ...

Two new approaches are presented to establish the existence of polytopal solutions to the discrete-data Lp Minkowski problem for all p > 1. As observed by Schneider [21], the Brunn-Minkowski theory springs from joining the notion of ordinary volume in Euclidean d-space, R, with that of Minkowski combinations of convex bodies. One of the cornerstones of the Brunn-Minkowski theory is the classica...

We prove a sharp stability result concerning how close homothetic sets attaining near-equality in the Brunn–Minkowski inequality are to being convex. In particular, resolving conjecture of Figalli and Jerison, we show there universal constants $\\smash{C_n,d_n>0}$ such that for $\\smash{A \\subset \\mathbb{R}^n}$ positive measure, if $\\smash{|\\frac{A+A}{2}\\setminus A| \\le d_n |A|}$, then $\...

The main purposes of this paper are to establish some new Brunn– Minkowski inequalities for width-integrals of mixed projection bodies and affine surface area of mixed bodies, together with their inverse forms.

We introduce a new operation between nonnegative integrable functions on R n $\mathbb {R}^n$ , that we call geometric combination; it is obtained via mass transportation approach, playing with inverse distribution functions. The main feature of this the Lebesgue integral combination equals mean two separate integrals; as natural consequence, derive functional inequality Prékopa–Leindler type. W...

In his Annals of Mathematics paper [2], Berndtsson proves an important result on the Nakano positivity holomorphic infinite-rank vector bundles whose fibers are Hilbert spaces consisting L2-functions with respect to a family weight functions {e−φ(t,⋅)}t∈U, varying in t∈U⊂Cm, over pseudoconvex domain. Using variant Hörmander's theorem due Donnelly and Fefferman, we show that Berndtsson's holds u...