Geometrical and Topological Properties of Bumps and Starlike Bodies in Banach Spaces

Abstract

While the topological and geometrical properties of convex bodies in Banach spaces are quite well understood (including their topological and smooth classification), much less is known about the structure of starlike bodies. Starlike bodies are important objects in nonlinear functional analysis as they appear as level sets of $n$-homogeneous polynomials on Banach spaces. Significant progress in the study of starlike bodies has been done in the last years by the efforts of Manuel Cepedello, Robert Deville, Tadeusz Dobrowolski, Marian Fabian and of the authors of the present survey. Its aim is to present these results in a coherent way, emphasizing the connections between infinite-dimensional topology and nonlinear functional analysis (such as the failure of Rolle's and Brouwer's theorems in infinite dimensions), and leading to new characterizations of smoothness properties of Banach spaces. One of the leading ideas of the paper is the use of bump functions as an instrument to study the properties of starlike bodies.\par The paper is divided into eight sections, their headings reflecting the contents and the organization of the paper: 1. Introduction; 2. Classifying starlike bodies; 3. Smooth Lipschitz contractibility of boundaries of starlike bodies in infinite dimensions; 4. The failure of Rolle's and Brouwer's theorems in infinite dimensions; 5. How small can the range of a derivative be? 6. How large does the range of a derivative look like? 8. Geometrical properties of starlike bodies. The failure of James' theorem for starlike bodies.\par Not all the proofs are given in full detail, the authors' emphasis being rather on the ideas lying behind them and on the connections between various properties and notions, avoiding cumbersome details. This survey is a valuable addition to the existing literature and can be used as a guide to this very active area of investigation in nonlinear functional analysis and infinite-dimensional topology.