For any bounded subset A of L^1 space, exists x fixed by every isometry of L^1 preserving A

You need to know: Banach space B with norm , linear map , bounded subset of B, space.

Background: Let B be a Banach space. A linear map is called an isometry if for all . For a subset , we say that f preserves A if for all . A point is called a fixed point of if .

The Theorem: On 7th December 2010, Uri Bader, Tsachik Gelander, and Nicolas Monod submitted to arxiv and Inventiones Mathematicae a paper in which they proved the following result. Let A be a non-empty bounded subset of an space B. Then there is a point in B fixed by every isometry of B preserving A. Moreover, one can choose a fixed point which minimises over all .

Short context: Starting with famous Banach’s Fixed Point Theorem, mathematicians developed fixed point theorems in various contexts, useful in different applications, see here for an example. The Theorem is a version of fixed point theorem for L1 spaces. The authors demonstrated that it has many applications. For example, it can be used to derive the optimal solution to so-called “derivation problem”, see the original paper for details.

Links: Free arxiv version of the original paper is here, journal version is here.

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