Fixed point theorems for metric spaces with a conical geodesic bicombing

We derive two fixed point theorems for a class of metric spaces that includes all Banach spaces and all complete Busemann spaces. We obtain our results by the use of a 1-Lipschitz barycenter construction and an existence result for invariant Radon probability measures. Furthermore, we construct a bounded complete Busemann space that admits an isometry without fixed points.Comment: 19 page

Almost minimal orthogonal projections

The projection constant $\Pi(E):=\Pi(E, \ell_\infty)$ of a finite-dimensional Banach space $E\subset\ell_\infty$ is by definition the smallest norm of a linear projection of $\ell_\infty$ onto $E$. Fix $n\geq 1$ and denote by $\Pi_n$ the maximal value of $\Pi(\cdot)$ amongst $n$-dimensional real Banach spaces. We prove for every $\varepsilon >0$ that there exist an integer $d\geq 1$ and an $n$-dimensional subspace $E\subset\ell_1^d$ such that $\Pi_n \leq \Pi(E, \ell_1^d) +2 \varepsilon$ and the orthogonal projection $P\colon \ell_1^d\to E$ is almost minimal in the sense that $\lVert P \rVert \leq \Pi(E, \ell_1^d)+\varepsilon$. As a consequence of our main result, we obtain a formula relating $\Pi_n$ to smallest absolute value row-sums of orthogonal projection matrices of rank $n$.Comment: final versio

Computation of maximal projection constants

The linear projection constant $\Pi(E)$ of a finite-dimensional real Banach space $E$ is the smallest number $C\in [0,+\infty)$ such that $E$ is a $C$-absolute retract in the category of real Banach spaces with bounded linear maps. We denote by $\Pi_n$ the maximal linear projection constant amongst $n$-dimensional Banach spaces. In this article, we prove that $\Pi_n$ may be determined by computing eigenvalues of certain two-graphs. From this result we obtain that the relative projection constants of codimension $n$ converge to $1+\Pi_n$. Furthermore, using the classification of $K_4$-free two-graphs, we give an alternative proof of $\Pi_2=\frac{4}{3}$. We also show by means of elementary functional analysis that for each integer $n\geq 1$ there exists a polyhedral $n$-dimensional Banach space $F_n$ such that $\Pi(F_n)=\Pi_n$.Comment: 27 page

Finsler currents

We propose a slight variant of Ambrosio and Kirchheim's definition of a metric current. We show that with this new definition it is possible to obtain certain volume functionals from Finsler geometry as mass measures of currents. As an application, we obtain a whole family of extendibly convex $n$-volume densities. This family includes the mass* and the circumscribed Riemannian volume densities.Comment: 21 page

On a problem of Mazur and Sternbach

We investigate Problem 155 form the "Scottish Book" due to S. Mazur and L. Sternbach. In modern terminology they asked if every bijective, locally isometric map between two real Banach spaces is always a global isometry. Recently, an affirmative answer when the source space is separable was obtained by M. Mori, using techniques related to the Mazur-Ulam theorem and its generalizations. The main purpose of this short note is to offer a different approach to Problem 155 motivated by recent advances in metric geometry. We show that it has an affirmative answer under the additional assumption that the map considered is a local isometry.Comment: revised version; we now only deal with local isometries, the general case remains ope

Extending and improving conical bicombings

We prove that any reversible conical bicombing can be extended to a conical bicombing on the injective hull of the underlying metric space. We also show that every proper metric space with a conical bicombing admits a consistent bicombing which satisfies certain convexity conditions. As an application, we prove that every group which acts geometrically on a proper metric space with a conical bicombing admits a $\mathcal{Z}$-structure.Comment: v2: Added section about Z-structures. Improved expositio

Approximating spaces of Nagata dimension zero by weighted trees

We prove that if a metric space $X$ has Nagata dimension zero with constant $c$, then there exists a dense subset of $X$ that is $8c$-bilipschitz equivalent to a weighted tree. The factor $8$ is the best possible if $c=1$, that is, if $X$ is an ultrametric space. This yields a new proof of a result of Chan, Xia, Konjevod and Richa. Moreover, as an application, we also obtain quantitative versions of certain metric embedding and Lipschitz extension results of Lang and Schlichenmaier. Finally, we prove a variant of our main theorem for $0$-hyperbolic proper metric spaces. This generalizes a result of Gupta.Comment: Revised version. We have corrected many minor inaccuracies in the tex

Filling minimality and Lipschitz-volume rigidity of convex bodies among integral current spaces

In this paper we consider metric fillings of convex bodies. We show that convex bodies $C\subset \mathbb{R}^n$ are the unique minimal fillings of their boundary metrics among all integral current spaces. To this end, we also prove that convex bodies enjoy the Lipschitz-volume rigidity property within the category of integral current spaces, which is well known in the smooth category. As a further application of this result, we answer a question of Perales concerning the intrinsic flat convergence of minimizing sequences for the Plateau problem.Comment: 25 pages, 1 figure, v3: we have added Corollary 1.3 concerning the LV-rigidity of the sphere. In version 2 we had already added Theorem 1.1, a filling minimality result for convex bodie

Geometric and analytic structures on metric spaces homeomorphic to a manifold

We study metric spaces homeomorphic to a closed oriented manifold from both geometric and analytic perspectives. We show that such spaces (which are sometimes called metric manifolds) admit a non-trivial integral current without boundary, provided they satisfy some weak assumptions. The existence of such an object should be thought of as an analytic analog of the fundamental class of the space and can also be interpreted as giving a way to make sense of Stokes' theorem in this setting. Using our existence result, we establish that Riemannian manifolds are Lipschitz-volume rigid among certain metric manifolds and we show the validity of (relative) isoperimetric inequalities in metric $n$-manifolds that are Ahlfors $n$-regular and linearly locally contractible. The former statement is a generalization of a well-known Lipschitz-volume rigidity result in Riemannian geometry and the latter yields a relatively short and conceptually simple proof of a deep theorem of Semmes about the validity of Poincar\'e inequalities in these spaces. Finally, as a further application, we also give sufficient conditions for a metric manifold to be rectifiable.Comment: Version 2: generalized and strengthened main existence results and added new results; added applications to Lipschitz-volume rigidit