On the convexity of Gaussian measures

Thursday, February 2, 2017 - 3:30pm
Math Sci 111
Ramon van Handel

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).

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