That the ball has the smallest surface area among all bodies of equal

volume was already known (it is said) to Dido, queen of Carthage. It was

realized by Minkowski 115 years ago that this isoperimetric property is a

manifestation of a much more general phenomenon: convexity of the Lebesgue

measure. The analogous isoperimetric and convexity properties of Gaussian

measures, which play a fundamental role in probability theory, are much

more recent discoveries due to Borell and Ehrhard. In particular, the

sharp convexity of Gaussian measures was only proved as recently as 2003,

and even its simplest properties remain poorly understood. In this talk, I

will introduce these phenomena that lie at the intersection of

probability, geometry, and analysis, and describe some of our recent

efforts to understand them better. In particular, I will discuss recent

work with Yair Shenfeld that settles the equality cases of the

Ehrhard-Borell inequalities using some unusual probabilistic and analytic

tools (such as hypoelliptic properties of degenerate parabolic operators).

# On the convexity of Gaussian measures

Date:

Thursday, February 2, 2017 - 3:30pm

Location:

Math Sci 111

Speaker:

Ramon van Handel

(Princeton)

Event Type: