Asymptotic Theory of Convex Bodies

Αριστεία ΙΙ - ATOCB - 3566

(KE): Γιαννόπουλος Απόστολος

This project is devoted to some fundamental problems of the asymptotic theory of convex bodies: the theory has its roots in geometric functional analysis and classical convexity and may be described as the quantitative study of the geometry of compact convex sets with non-empty interior in the n-dimensional Euclidean space, as the dimension n grows to infinity.

Understanding the typical behaviour of high-dimensional objects lies at the heart of asymptotic geometric analysis, a much broader interdisciplinary area with applications to probability theory, statistical physics, algorithmic geometry, combinatorics and complexity.

We focus on the study of the distribution of volume on high-dimensional convex bodies and on well-known problems about the geometry of the Banach-Mazur compactum, using tools from geometric analysis, probability theory and combinatorics. Our main directions of research are:

• Asymptotic concentration of measure on isotropic convex bodies and the hyperplane conjecture.
  
• Geometry of log-concave probability measures.
  
• Asymptotic shape of random polytopes.
  
• Estimates on Banach-Mazur distances and the geometry of the n-dimensional compactum.