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Register a new userReverse Triangle Inequalities for Potentials
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We study the reverse triangle inequalities for suprema of logarithmic potentials on compact sets of the plane. This research is motivated by the inequalities for products of supremum norms of polynomials. We find sharp additive constants in the inequalities for potentials, and give applications of our results to the generalized polynomials. We also obtain sharp inequalities for products of norms of the weighted polynomials $w^nP_n, deg(P_n)le n,$ and for sums of suprema of potentials with external fields. An important part of our work in the weighted case is a Riesz decomposition for the weighted farthest-point distance function.
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We study reverse triangle inequalities for Riesz potentials and their connection with polarization. This work generalizes inequalities for sup norms of products of polynomials, and reverse triangle inequalities for logarithmic potentials. The main tool used in the proofs is the representation for a power of the farthest distance function as a Riesz potential of a unit Borel measure.
The Alexandrov--Fenchel inequality bounds from below the square of the mixed volume $V(K_1,K_2,K_3,ldots,K_n)$ of convex bodies $K_1,ldots,K_n$ in $mathbb{R}^n$ by the product of the mixed volumes $V(K_1,K_1,K_3,ldots,K_n)$ and $V(K_2,K_2,K_3,ldots,K_n)$. As a consequence, for integers $alpha_1,ldots,alpha_minmathbb{N}$ with $alpha_1+cdots+alpha_m=n$ the product $V_n(K_1)^{frac{alpha_1}{n}}cdots V_n(K_m)^{frac{alpha_m}{n}} $ of suitable powers of the volumes $V_n(K_i)$ of the convex bodies $K_i$, $i=1,ldots,m$, is a lower bound for the mixed volume $V(K_1[alpha_1],ldots,K_m[alpha_m])$, where $alpha_i$ is the multiplicity with which $K_i$ appears in the mixed volume. It has been conjectured by Ulrich Betke and Wolfgang Weil that there is a reverse inequality, that is, a sharp upper bound for the mixed volume $V(K_1[alpha_1],ldots,K_m[alpha_m])$ in terms of the product of the intrinsic volumes $V_{alpha_i}(K_i)$, for $i=1,ldots,m$. The case where $m=2$, $alpha_1=1$, $alpha_2=n-1$ has recently been settled by the present authors (2020). The case where $m=3$, $alpha_1=alpha_2=1$, $alpha_3=n-2$ has been treated by Artstein-Avidan, Florentin, Ostrover (2014) under the assumption that $K_2$ is a zonoid and $K_3$ is the Euclidean unit ball. The case where $alpha_2=cdots=alpha_m=1$, $K_1$ is the unit ball and $K_2,ldots,K_m$ are zonoids has been considered by Hug, Schneider (2011). Here we substantially generalize these previous contributions, in cases where most of the bodies are zonoids, and thus we provide further evidence supporting the conjectured reverse Alexandrov--Fenchel inequality. The equality cases in all considered inequalities are characterized. More generally, stronger stability results are established as well.
New upper bounds on the relative entropy are derived as a function of the total variation distance. One bound refines an inequality by Verd{u} for general probability measures. A second bound improves the tightness of an inequality by Csisz{a}r and Talata for arbitrary probability measures that are defined on a common finite set. The latter result is further extended, for probability measures on a finite set, leading to an upper bound on the R{e}nyi divergence of an arbitrary non-negative order (including $infty$) as a function of the total variation distance. Another lower bound by Verd{u} on the total variation distance, expressed in terms of the distribution of the relative information, is tightened and it is attained under some conditions. The effect of these improvements is exemplified.
This paper is devoted to a new family of reverse Hardy-Littlewood-Sobolev inequalities which involve a power law kernel with positive exponent. We investigate the range of the admissible parameters and characterize the optimal functions. A striking open question is the possibility of concentration which is analyzed and related with nonlinear diffusion equations involving mean field drifts.
In this expository article we introduce a diagrammatic scheme to represent reverse classes of weights and some of their properties.
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