arXiv research

Given $L$ a convex body, the $L_p$-Busemann Random Simplex Inequality is closely related to the centroid body $Γ_p L$ for $p=1$ and $2$, and only in these cases it can be proved using the $L_p$-Busemann-Petty centroid inequality. We define a convex body $N_p L$ and prove an isoperimetric inequality for $(N_p L)^\circ$ …

The paper extends Busemann's inequalities to complex and quaternionic spaces.

New optimal isosystolic inequality found for Finsler reversible 2-tori.

Sharp Lp affine isoperimetric inequalities are established for the entire class of Lp projection bodies and the entire class of Lp centroid bodies. These new inequalities strengthen the Lp Petty projection and the Lp Busemann--Petty centroid inequality.

The paper defines subdifferentials on Hadamard manifolds and identifies conditions for Fenchel conjugate equality.

In [Centro-affine invariants for smooth convex bodies, Int. Math. Res. Notices. doi: 10.1093/imrn/rnr110, 2011] Stancu introduced a family of centro-affine normal flows, $p$-flow, for $1\leq p<\infty.$ Here we investigate the asymptotic behavior of the planar $p$-flow for $p=\infty$ in the class of smooth, origin-symme…

Smooth Busemann functions found in harmonic Finsler spaces.

Study Busemann spaces with measures under MCP, proving rigidity and structure theorems.

Decomposes Busemann spaces into simpler structures.

We present short proofs of all known topological properties of general Busemann $G$-spaces (at present no other property is known for dimensions more than four). We prove that all small metric spheres in locally $G$-homogeneous Busemann $G$-spaces are homeomorphic and strongly topologically homogeneous. This is a key r…

Study calculates Hessian of Busemann function on Damek-Ricci spaces.

First example of open manifold with positive Ricci curvature and non-proper Busemann function.

We introduce a new volume definition on normed vector spaces. We show that the induced $k$-area functionals are convex for all $k$. In the particular case $k=2$, our theorem implies that Busemann's 2-volume density is convex, which was recently shown by Burago-Ivanov. We also show how the new volume definition is relat…

Busemann G-spaces with Finsler metrics

Busemann points are sparse in Teichmüller spaces.

This is a biography of Herbert Busemann (1905--1994). The final version will appear in Volume I of the Selected Works of Herbert Busemann (2 volumes, Springer Verlag, to appear in 2017).

We explore a connection between the Finslerian area functional based on the Busemann-Hausdorff-volume form, and well-investigated Cartan functionals to solve Plateau's problem in Finsler 3-space, and prove higher regularity of solutions. Free and semi-free geometric boundary value problems, as well as the Douglas probl…

In this article we show that every geodesic is rank one and the Hessian of Busemann functions is positive definite for a harmonic Damek-Ricci space, a two step solvable Lie group with a left invariant metric. Moreover, the eigenspace of the Hessian of Busemann functions on a Hadamard manifold $(M,g)$ corresponding to e…

We determine the asymptotic behaviour of extremal length along arbitrary Teichmüller rays. This allows us to calculate the endpoint in the Gardiner-Masur boundary of any Teichmüller ray. We give a proof that this compactification is the same as the horofunction compactification. An important subset of the latter is the…

The paper extends inequalities for convex bodies to higher dimensions and various norms.

The study proves properties of intersections of horospheres in harmonic spaces.

Each compact Riemannian manifold with no conjugate points admits a family of functions whose integrals vanish exactly when central Busemann functions split linearly. These functions vanish when all central Busemann functions are sub- or superharmonic. When central Busemann functions are convex or concave, they must be …

We prove that every Riemannian metric on the 2-disc such that all its geodesics are minimal, is a minimal filling of its boundary (within the class of fillings homeomorphic to the disc). This improves an earlier result of the author by removing the assumption that the boundary is convex. More generally, we prove this r…

This paper is a commentary and a reading guide to three papers by Herbert Busemann, Über die Geometrien, in denen die "Kreise mit unendlichem Radius" die kürzesten Linien sind." (On the geometries where circles of infinite radius are the shortest lines) (1932), "Paschsches Axiom und Zweidimensionalität," (Pasch's Axiom…

We discuss two different in general natural approaches to the ideal closure and ideal boundary of Busemann nonpositively curved metric space. It is shown that the identity map of the space admits surjective continuation from its coarse ideal closure to the weak one. We consider some situations when these closures coinc…

Researchers find Busemann functions in Wasserstein space, enabling efficient projections and distances.

The paper explores isometric models and Busemann functions for Funk and Hilbert discs.

In this paper, we shall show that the metric boundary of the Teichmueller space with respect to the Teichmueller distance contains non-Busemann points when the complex dimension of the Teichmueller space is at least two.

A new depth measure and median defined on Hadamard manifolds.

We determine the set of Busemann points of an arbitrary finite-dimensional normed space. These are the points of the horofunction boundary that are the limits of "almost-geodesics". We prove that all points in the horofunction boundary are Busemann points if and only if the set of extreme sets of the dual unit ball is …

In the first part Busemann concavity as non-negative curvature is introduced and a bi-Lipschitz splitting theorem is shown. Furthermore, if the Hausdorff measure of a Busemann concave space is non-trivial then the space is doubling and satisfies a Poincaré condition and the measure contraction property. Using a compari…

Researchers create metrics on hyperbolic space's tangent bundle.

Given a Finsler manifold $(M,F)$, it is proved that the first eigenvalue of the Finslerian $p$-Laplacian is bounded above by a constant depending on $\ p$, the dimension of $M$, the Busemann-Hausdorff volume and the reversibility constant of $(M,F)$. For a Randers manifold $(M,F:=\sqrt{g}+β)$, where $g$ is a Riemannian…

We present two classical conjectures concerning the characterization of manifolds: the Bing Borsuk Conjecture asserts that every $n$-dimensional homogeneous ANR is a topological $n$-manifold, whereas the Busemann Conjecture asserts that every $n$-dimensional $G$-space is a topological $n$-manifold. The key object in bo…

Hilbert's fourth problem asks for the construction and the study of metrics on subsets of projective space for which the projective line segments are geodesics. Several solutions of the problem were given so far, depending on more precise interpretations of this problem, with various additional conditions satisfied. Th…

We prove that a Finsler metric is nonpositively curved in the sense of Busemann if and only if it is affinely equivalent to a Riemannian metric of nonpositive sectional curvature. In other terms, such Finsler metrics are precisely Berwald metrics of nonpositive flag curvature. In particular in dimension 2 every such me…

A new learning method using hyperbolic geometry for class labels.

We study rays and co-rays in the Wasserstein space $P_p(\mathcal{X})$ ($p > 1$) whose ambient space $\mathcal{X}$ is a complete, separable, non-compact, locally compact length space. We show that rays in the Wasserstein space can be represented as probability measures concentrated on the set of rays in the ambient spac…

Study weakly weighted Einstein-Finsler metrics, showing specific curvature properties and characterizing them.

In this paper known results of symmetric orthogonality, as introduced by G. Birkhoff, and non-expansive nearest point projections are extended from the linear to the metric setting. If the space has non-positive curvature in the sense Busemann then it is shown that those concepts are actually equivalent. In the end it …

Paper extends tree bijection for hyperbolic surfaces without requiring cusps.

The horoboundary of Teichmüller space is path connected and has non-dense Busemann points.

Recently, the old notion of causal boundary for a spacetime V has been redefined in a consistent way. The computation of this boundary $\partial V$ for a standard conformally stationary spacetime V = R x M, suggests a natural compactification $M_B$ associated to any Riemannian metric on M or, more generally, to any Fin…

We show that the co-rays to a ray in a complete non-compact Finsler manifold contain geodesic segments to upper level sets of Busemann functions. Moreover, we characterise the co-point set to a ray as the cut locus of such level sets. The structure theorem of the co-point set on a surface, namely that is a local tree, …

Extends topological results for nonpositive curvature spaces.

We investigate the finiteness structure of a complete non-compact $n$-dimensional Riemannian manifold $M$ whose radial curvature at a base point of $M$ is bounded from below by that of a non-compact von Mangoldt surface of revolution with its total curvature greater than $π$. We show, as our main theorem, that all Buse…

New theorem splits weighted Lorentz-Finsler manifolds into simpler parts.

We show that a non-compact (forward) complete Finsler manifold whose Holmes- Thompson volume is infinite admits no non-trivial convex functions. We apply this result to some Finsler manifolds whose Busemann function is convex.