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Series: High Dimensional Seminar

TBA

Series: High Dimensional Seminar

We discuss the asymptotic value of the maximal perimeter of a convex set in an n-dimensional space with respect to certain classes of measures. Firstly, we derive a lower bound for this quantity for an arbitrary probability measure with first two moments bounded; the lower bound depends on the moments only. This lower bound is sharp in the case of the Gaussian measure (as was shown by Nazarov in 2001), and, more generally, in the case of rotation invariant log-concave measures (as was shown by the author in 2014). We show, that this lower bound is also sharp for a class of smooth log-concave measures satisfying certain uniform bounds on the hessian of the potential. In addition, we show a uniform upper bound of Cn for all isotropic log-concave measures, which is attained for the uniform distribution on the cube. Some improved bounds are also obtained for the Poisson density.

Series: High Dimensional Seminar

TBA

Series: High Dimensional Seminar

TBA

Series: High Dimensional Seminar

Series: High Dimensional Seminar

TBA

Series: High Dimensional Seminar

Series: High Dimensional Seminar

TBA

Series: High Dimensional Seminar

A single soap bubble has a spherical shape since it minimizes its surface area subject to a fixed enclosed volume of air. When two soap bubbles collide, they form a “double-bubble” composed of three spherical caps. The double-bubble minimizes total surface area among all sets enclosing two fixed volumes. This was proven mathematically in a landmark result by Hutchings-Morgan-Ritore-Ros and Reichardt using the calculus of variations in the early 2000s. The analogous case of three or more Euclidean sets is considered difficult if not impossible. However, if we replace Lebesgue measure in these problems with the Gaussian measure, then recent work of myself (for 3 sets) and of Milman-Neeman (for any number of sets) can actually solve these problems. We also use the calculus of variations. Time permitting, we will discuss an improvement to the Milman-Neeman result and applications to optimal clustering of data and to designing elections that are resilient to hacking. <a href="http://arxiv.org/abs/1901.03934">http://arxiv.org/abs/1901.03934</a>

Series: High Dimensional Seminar

Moment problem is a classical question in real analysis, which asks whether a set of moments can be realized as integration of corresponding monomials with respect to a Borel measure. Truncated moment problem asks the same question given a finite set of moments. I will explain how some of the fundamental results in the truncated moment problem can be proved (in a very general setting) using elementary convex geometry. No familiarity with moment problems will be assumed. This is joint work with Larry Fialkow.