BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:James Melbourne (CIMAT) DTSTART:20220131T200000Z DTEND:20220131T210000Z DTSTAMP:20260423T021347Z UID:paw/55 DESCRIPTION:Title: On a reversal of Lyapunov's inequality for log-concave sequences\nby Jame s Melbourne (CIMAT) as part of Probability and Analysis Webinar\n\n\nAbstr act\nLog-concave sequences appear naturally in a variety of fields. For ex ample in convex geometry the Alexandrov-Fenchel inequalities demonstrate t he intrinsic volumes of a convex body to be log-concave\, while in combina torics the resolution of the Mason conjecture shows that the number of ind ependent 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 me asure spaces satisfying concavity conditions are known to satisfy a sort o f concavity reversal of both Lyapunov's inequality\, due to Borell\, while the Prekopa-Leindler inequality gives a reversal of Holder. These inequa lities are foundational in convex geometry\, give Renyi entropy comparison s in information theory\, the Gaussian log-Sobolev inequality\, and more g enerally the HWI inequality in optimal transport among other applications. An analogous theory has been developing in the discrete setting. In thi s talk we establish a reversal of Lyapunov's inequality for monotone log-c oncave sequences\, settling a conjecture of Havrilla-Tkocz and Melbourne-T kocz. A strengthened version of the same conjecture is disproved through c ounter-examples.\n LOCATION:https://researchseminars.org/talk/paw/55/ END:VEVENT END:VCALENDAR