Fixed point and Lipschitz extension theorems for barycentric metric spaces

Abstract

The subject of this doctoral thesis is the class of barycentric metric spaces, which encompasses both Banach spaces and complete CAT(0) spaces. Encouraged by known results as well as open questions in the context of CAT(0) spaces, we study similar objectives in the framework of barycentric metric spaces. For example, we show that certain fixed point properties, which are given in CAT(0) spaces, do not hold for some barycentric metric spaces, and prove two fixed point results adapted to the new situation. These results are phrased for the class of metric spaces that allow a conical bicombing; this is no restriction, since the class of barycentric metric spaces agrees with this class. This equality leads to a variety of questions regarding the existence and uniqueness of certain classes of conical bicombings. In particular, we consider conical bicombings on open subsets of normed vector spaces and show that these bicombings are locally given by linear segments. This result implies that any open convex subset in a large class of Banach spaces possesses a unique consistent conical bicombing. Besides this, we consider various Lipschitz extension problems, where in some cases any complete barycentric metric space may appear as target space. One such Lipschitz extension problem involves the extension of a Lipschitz function to finitely many additional points. Our contribution consists of finding upper bounds for the distortion of the Lipschitz constant, and we construct examples which demonstrate that we found the best possible bounds in the case of an extension to one additional point. Many Lipschitz extension constants may be computed by solving an associated linear extension problem, which is why, in the last part, we turn our attention to absolute linear projection constants of real Banach spaces. We succeeded in finding a formula for the maximal linear projection constant amongst \(n\)-dimensional Banach spaces. By means of this formula, we give another proof of the Grünbaum conjecture, which was first proven by Chalmers and Lewicki in 2010.