# Isometric Embedding of Busemann Surfaces into $$L_1$$L1

@article{Chalopin2015IsometricEO, title={Isometric Embedding of Busemann Surfaces into \$\$L\_1\$\$L1}, author={J{\'e}r{\'e}mie Chalopin and Victor Chepoi and Guyslain Naves}, journal={Discrete \& Computational Geometry}, year={2015}, volume={53}, pages={16-37} }

In this paper, we prove that any non-positively curved 2-dimensional surface (alias, Busemann surface) is isometrically embeddable into $$L_1$$L1. As a corollary, we obtain that all planar graphs which are 1-skeletons of planar non-positively curved complexes with regular Euclidean polygons as cells are $$L_1$$L1-embeddable with distortion at most $$2$$2. Our results significantly improve and simplify the results of the recent paper by A. Sidiropoulos (Non-positive curvature and the planar… Expand

#### Paper Mentions

#### 2 Citations

Packing and Covering with Balls on Busemann Surfaces

- Mathematics, Computer Science
- Discret. Comput. Geom.
- 2017

It is proved that for any compact subset S of a Busemann surface S,d, and any positive numberδ, the minimum number of closed balls of radiusδ with centers at S is at most 19 times the maximum number of disjoint closed Balls of radius. Expand

A new approach to low-distortion embeddings of finite metric spaces into non-superreflexive Banach spaces

- Mathematics
- 2016

The main goal of this paper is to develop a new embedding method which we use to show that some finite metric spaces admit low-distortion embeddings into all non-superreflexive spaces. This method is… Expand

#### References

SHOWING 1-10 OF 30 REFERENCES

Non-positive Curvature and the Planar Embedding Conjecture

- Mathematics, Computer Science
- 2013 IEEE 54th Annual Symposium on Foundations of Computer Science
- 2013

It is shown that every planar metric of non-positive curvature admits a constant-distortion embedding into L1, which confirms the planar embedding conjecture for the case ofnon-positively curved metrics. Expand

Curvature and Geometry of Tessellating Plane Graphs

- Mathematics, Computer Science
- Discret. Comput. Geom.
- 2001

We show that the growth of plane tessellations and their edge graphs may be controlled from below by upper bounds for the combinatorial curvature. Under the assumption that every geodesic path may be… Expand

Convexity in Topological Affine Planes

- Mathematics, Computer Science
- Discret. Comput. Geom.
- 2007

We extend to topological affine planes the standard theorems of convexity, among them the separation theorem, the anti-exchange theorem, Radon's, Helly's, Caratheodory's, and Kirchberger's theorems,… Expand

Metric Spaces, Convexity and Nonpositive Curvature

- Mathematics
- 2004

This is the second edition of a book which appeared in 2005. The new edition is an expanded and revised version. The book is about metric spaces of nonpositive curvature in the sense of Busemann,… Expand

Lectures on discrete geometry

- Computer Science, Mathematics
- Graduate texts in mathematics
- 2002

This book is primarily a textbook introduction to various areas of discrete geometry, in which several key results and methods are explained, in an accessible and concrete manner, in each area. Expand

On lipschitz embedding of finite metric spaces in Hilbert space

- Mathematics
- 1985

It is shown that anyn point metric space is up to logn lipeomorphic to a subset of Hilbert space. We also exhibit an example of ann point metric space which cannot be embedded in Hilbert space with… Expand

Surveys on discrete and computational geometry : twenty years later : AMS-IMS-SIAM Joint Summer Research Conference, June 18-22, 2006, Snowbird, Utah

- Mathematics
- 2008

Musings on discrete geometry and ""20 years of discrete & computational geometry"" by B. Grunbaum State of the union (of geometric objects) by P. K. Agarwal, J. Pach, and M. Sharir Metric graph… Expand

Metric graph theory and geometry: a survey

- 2006

The article surveys structural characterizations of several graph classes defined by distance properties, which have in part a general algebraic flavor and can be interpreted as subdirect… Expand

Multicommodity flows in planar graphs

- Computer Science, Mathematics
- J. Comb. Theory, Ser. B
- 1981

This paper solves the problem of when is there a flow for each i, between s i and t i and of value q i, such that the total flow through each edge does not exceed its capacity. Expand

Nonpositive Curvature and the Ptolemy Inequality

- Mathematics
- 2007

We provide examples of nonlocally, compact, geodesic Ptolemy metric spaces which are not uniquely geodesic. On the other hand, we show that locally, compact, geodesic Ptolemy metric spaces are… Expand