On the mean width of log-concave functions

In this work we present a new, natural, definition for the mean width of log-concave functions. We show that the new definition coincide with a previous one by B. Klartag and V. Milman, and deduce some properties of the mean width, including an Urysohn type inequality. Finally, we prove a functional version of the finite volume ratio estimate and the low-M* estimate.Comment: 15 page

Dynamic Disappointment Aversion: Don't Tell Me Anything Until You Know For Sure

We show that for a disappointment-averse decision maker, splitting a lottery into several stages reduces its value. To do this, we extend Gul.s (1991) model of disappointment aversion into a dynamic setting while keeping its basic characteristics intact. The result depends solely on the sign of the coefficient of disappointment aversion. It can help explain why people often buy periodic insurance for moderately priced objects, such as electrical appliances and cellular phones, at much more than the actuarially fair rate.Disappointment aversion, recursive preferences, compound lotteries

A direct proof of the functional Santalo inequality

We give a simple proof of a functional version of the Blaschke-Santalo inequality due to Artstein, Klartag and Milman. The proof is by induction on the dimension and does not use the Blaschke-Santalo inequality.Comment: 4 pages, file might be slighlty different from the published versio

Analysis of polarity

We develop a differential theory for the polarity transform parallel to that for the Legendre transform, which is applicable when the functions studied are "geometric convex", namely convex, non-negative and vanish at the origin. This analysis may be used to solve a family of first order equations reminiscent of Hamilton--Jacobi and conservation law equations, as well as some second order Monge-Ampere type equations. A special case of the latter, that we refer to as the homogeneous polar Monge--Ampere equation, gives rise to a canonical method of interpolating between convex functions