On a reversal of Lyapunov's inequality for log-concave sequences

Abstract: Log-concave sequences appear naturally in a variety of fields. For example in convex geometry the Alexandrov-Fenchel inequalities demonstrate the intrinsic volumes of a convex body to be log-concave, while in combinatorics the resolution of the Mason conjecture shows that the number of independent sets of size n in a matroid form a log-concave sequence as well. By Lyapunov's inequality we refer to the log-convexity of the (p-th power) of the L^p norm of a function with respect to an arbitrary measure, an immediate consequence of Holder's inequality. In the continuous setting measure spaces satisfying concavity conditions are known to satisfy a sort of concavity reversal of both Lyapunov's inequality, due to Borell, while the Prekopa-Leindler inequality gives a reversal of Holder. These inequalities are foundational in convex geometry, give Renyi entropy comparisons in information theory, the Gaussian log-Sobolev inequality, and more generally the HWI inequality in optimal transport among other applications. An analogous theory has been developing in the discrete setting. In this talk we establish a reversal of Lyapunov's inequality for monotone log-concave sequences, settling a conjecture of Havrilla-Tkocz and Melbourne-Tkocz. A strengthened version of the same conjecture is disproved through counter-examples.

mathematical physicsanalysis of PDEsclassical analysis and ODEscombinatoricscomplex variablesfunctional analysisinformation theorymetric geometryoptimization and controlprobability

Audience: researchers in the topic

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