Curvature flows and convex geometry

这个程序的目的是研究曲率流在凸和仿射几何中的应用。它的主要部分涉及凸超曲面的高斯曲率的某些权力的流动。从本质上讲，这些流动将被研究，以了解如何超曲面演变的时间，以一个更优的形状，并获得这些渐近形状的重要信息。其中一个著名的问题，出现在Brunn-Minkowski-Firey理论的凸体推广的经典Minkowski问题，找到凸体与规定的曲率函数。在经典问题的推广中，人们寻求具有给定L_p曲率函数的凸体，这个问题由于在信息论等领域的进一步应用而引起了人们的广泛关注。过去的工作，我们所做的情况下的广义问题，导致结果的存在性和唯一性，这样的凸体通过曲率流的渐近行为。其中一个关键任务是改进上述流程的方法，以解决一类其他更困难的情况。项目中还有一些其他几何问题。他们集中沿着两个主要方向：研究新的仿射不变量的凸体和问题的充分条件，为两个有界凸域在欧几里德平面或空间，使一个域包含一个全等副本的其他。尽管他们明显的多样性，我们的目标作为一个共同的线程使用的几何发展方程。各种技术来自偏微分方程，凸和微分几何是必要的，以接近这些主题，每一个本身是一个具有挑战性的和重要的问题，在该领域。

The purpose of this program is to study applications of curvature flows to convex and affine geometry. Its main part concerns flows of convex hypersurfaces by certain powers of the Gauss curvature. In essence, these flows will be studied to understand how a hypersurface evolves in time to a more optimal shape and to get significant information on these asymptotic shapes.One of the famous questions that arises within the Brunn-Minkowski-Firey theory of convex bodies generalizes the classical Minkowski problem of finding convex bodies with prescribed curvature function. In the extension of the classical problem, one seeks convex bodies with prescribed L_p curvature function, a question which generated a lot attention in view of its further applications in other areas like information theory. Past work that we did on a case of the generalized problem has led to results on the existence and uniqueness of such convex bodies via the asymptotic behavior of a curvature flow. One of the key tasks is to refine this method for the aforementioned flows to solve a class of other, more difficult, cases.There are also a number of other geometric questions in the project. They concentrate along two main directions: the study of new affine invariants of convex bodies and the question of finding sufficient conditions for two bounded convex domains in the Euclidean plane or space such that one domain contains a congruent copy of the other.Despite their apparent diversity, our objectives have as a common thread the use of geometric evolution equations. A variety of techniques coming from PDE, convex and differential geometry are necessary to approach these topics, each in itself a challenging and important problem in the field.