# An Invitation to Tropical Alexandrov Curvature

@inproceedings{Amendola2021AnIT, title={An Invitation to Tropical Alexandrov Curvature}, author={Carlos Am'endola and Anthea Monod}, year={2021} }

We study Alexandrov curvature in the tropical projective torus with respect to the tropical metric. Alexandrov curvature is a generalization of classical Riemannian sectional curvature to more general metric spaces; it is determined by a comparison of triangles in an arbitrary metric space to corresponding triangles in Euclidean space. In the polyhedral setting of tropical geometry, triangles are a combinatorial object, which adds a combinatorial dimension to our analysis. We study the effect…

## Figures and Tables from this paper

## References

SHOWING 1-10 OF 65 REFERENCES

On the total curvature of tropical hypersurfaces

- Mathematics
- 2013

This paper studies the curvatures of amoebas and real amoebas (i.e. essentially logarithmic curvatures of the complex and real parts of a real algebraic hypersurface) and of tropical and real…

A visual introduction to Riemannian curvatures and some discrete generalizations

- Mathematics
- 2012

We try to provide a visual introduction to some objects used in Riemannian geometry: parallel transport, sectional curvature, Ricci curvature, Bianchi identities... We then explain some of the…

The tropical Grassmannian

- Mathematics
- 2003

In tropical algebraic geometry, the solution sets of polynomial equations are piecewise-linear. We introduce the tropical variety of a polynomial ideal, and we identify it with a polyhedral…

A Note on Tropical Triangles in the Plane

- Mathematics
- 2009

We define transversal tropical triangles (affine and projective) and characterize them via six inequalities to be satisfied by the coordinates of the vertices. We prove that the vertices of a…

Metric Spaces of Non-Positive Curvature

- Mathematics
- 1999

This book describes the global properties of simply-connected spaces that are non-positively curved in the sense of A. D. Alexandrov, and the structure of groups which act on such spaces by…

Tropical optimal transport and Wasserstein distances

- MathematicsInformation Geometry
- 2021

We study the problem of optimal transport in tropical geometry and define the Wasserstein- p distances in the continuous metric measure space setting of the tropical projective torus. We specify the…

Convexity in Tree Spaces

- MathematicsSIAM J. Discret. Math.
- 2017

The geometry of metrics and convexity structures on the space of phylogenetic trees is studied, which is here realized as the tropical linear space of all ultrametrics and the tropical metric arises from the theory of orthant spaces.

CRITICAL POINTS AND CURVATURE FOR EMBEDDED POLYHEDRA

- Mathematics
- 1967

Recently a new insight into the Gauss-Bonnet Theorem and other problems in global differential geometry has come about through the connection between total curvature of embedded smooth manifolds and…

Barycenters in Alexandrov spaces of curvature bounded below

- Mathematics
- 2012

We investigate barycenters of probability measures on proper Alexandrov spaces of curvature bounded below, and show that they enjoy several properties relevant to or different from those in metric…