James Melbourne




Monday, January 31, 2022 - 12:00pm


Zoom: Meeting ID: 961 2738 3116 Passcode: the last four digits above in the reverse order

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