Isoperimetric Games

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. This isoperimetric property follows from the fact that Lebesgue measure is log-concave. The analogous statement for Gaussian distributions - that half-spaces have the smallest Gaussian surface area among all sets of the same probability -plays an important role in high-dimensional probability. However, Gaussian isoperimetry does not follow from log-concavity of the Gaussian distribution, but from a much more delicate convexity property of Gaussian measures known as Ehrhard's inequality. My aim in this talk is to exhibit an unexpected link between Ehrhard's inequality and game theory: the convexity of Gaussian measures arises as the result of a remarkable game between two players who compete for control of a Brownian motion. This approach makes it possible to derive new geometric inequalities for Gaussian measures.