In the context of his work on maximal functions in the 1980s, Jean Bourgain came across the following geometric question: Is there c > 0 such that for any dimension n and any convex body K ⊆ Rn of volume one, there exists a hyperplane H such that the (n-1)-dimensional volume of K ∩ H is at least c? This innocent and seemingly obvious question (which remains unanswered!) has established a new direction in high-dimensional geometry. It has emerged as an "engine" that inspired the discovery of many deep results and unexpected connections. Here we provide a survey of these developments, including many of Bourgain's results.
|Title of host publication
|Analysis at Large
|Subtitle of host publication
|Dedicated to the Life and Work of Jean Bourgain
|Springer International Publishing
|Number of pages
|Published - 1 Nov 2022