Mathematical Sciences: Stability Problems In The Theory Of Convex Sets

Helmut Groemer将研究与凸集相关的稳定性问题。凸集理论中的许多定理表明，在某些条件下，一个物体具有重要的特殊类型，或者两个物体以特定的方式相互关联。与这样的结果相对应，通常很容易提出相关的稳定性问题。例如，有人可能会问，根据等周差，一个物体离球有多近。研究者已经在这方面取得了一些进展，特别是在与凸体投影有关的结果方面。他未来的研究将集中在与布伦-闵可夫斯基定理、对称过程、内切体和外切体相关的稳定性问题上。预期的技术将涉及球面谐波和氡变换的几何适应。

Helmut Groemer will carry out research into stability problems associated with convex sets. Many theorems in the theory of convex sets state that under certain conditions a body is of an important special type or that two bodies are related to one another in a particular way. Corresponding to such results it is often easy to pose an associated stability problem. For example one might ask how close a body is to being a ball in terms of its isoperimetric deficit. The investigator has already made some progress in this direction notably in the case of results connected with projections of convex bodies. His future research will focus on stability problems associated with the Brunn-Minkowski theorem, symmetrization procedures, inscribed and circumscribed bodies. The anticipated techniques will involve geometric adaptations of spherical harmonics and Radon transforms.