# On the equality case in Ehrhart's volume conjecture

@article{Nill2012OnTE, title={On the equality case in Ehrhart's volume conjecture}, author={Benjamin Nill and A. Paffenholz}, journal={arXiv: Combinatorics}, year={2012} }

Ehrhart's conjecture proposes a sharp upper bound on the volume of a convex body whose barycenter is its only interior lattice point. Recently, Berman and Berndtsson proved this conjecture for a class of rational polytopes including reflexive polytopes. In particular, they showed that the complex projective space has the maximal anticanonical degree among all toric Kaehler-Einstein Fano manifolds. In this note, we prove that projective space is the only such toric manifold with maximal degree… Expand

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#### References

SHOWING 1-10 OF 23 REFERENCES

The volume of K\"ahler-Einstein Fano varieties and convex bodies

- Mathematics
- 2012

We show that the complex projective space has maximal degree (volume) among all n-dimensional Kahler-Einstein Fano manifolds admitting a holomorphic C^*-action with a finite number of fixed points.… Expand

Greatest lower bounds on the Ricci curvature of Fano manifolds

- Mathematics
- Compositio Mathematica
- 2010

Abstract On a Fano manifold M we study the supremum of the possible t such that there is a Kähler metric ω∈c1(M) with Ricci curvature bounded below by t. This is shown to be the same as the maximum… Expand

Examples of Kähler–Einstein toric Fano manifolds associated to non-symmetric reflexive polytopes

- Mathematics
- 2011

In this note we report on examples of 7- and 8-dimensional toric Fano manifolds whose associated reflexive polytopes are not symmetric, but they still admit a Kähler–Einstein metric. This answers a… Expand

Introduction to Toric Varieties.

- Mathematics
- 1993

Toric varieties are algebraic varieties arising from elementary geometric and combinatorial objects such as convex polytopes in Euclidean space with vertices on lattice points. Since many algebraic… Expand

Examples of non-symmetric K\"ahler-Einstein toric Fano manifolds

- Mathematics, Physics
- 2009

In this note we report on examples of 7- and 8-dimensional toric Fano manifolds that are not symmetric and still admit a Kaehler-Einstein metric. This answers a question first posed by V.V. Batyrev… Expand

Gorenstein toric Fano varieties

- Mathematics
- 2005

Abstract.We investigate Gorenstein toric Fano varieties by combinatorial methods using the notion of a reflexive polytope which appeared in connection to mirror symmetry. The paper contains… Expand

Complete toric varieties with reductive automorphism group

- Mathematics
- 2006

We give equivalent and sufficient criteria for the automorphism group of a complete toric variety, respectively a Gorenstein toric Fano variety, to be reductive. In particular we show that the… Expand

Complete classification of reflexive polyhedra in four dimensions

- Physics, Mathematics
- 2000

Four dimensional reflexive polyhedra encode the data for smooth Calabi-Yau threefolds that are hypersurfaces in toric varieties, and have important applications both in perturbative and in… Expand

Greatest lower bounds on Ricci curvature for toric Fano manifolds

- Mathematics
- 2009

Abstract In this short note, based on the work of Wang and Zhu (2004) [8] , we determine the greatest lower bounds on Ricci curvature for all toric Fano manifolds.

On measures of symmetry and floating bodies

- Mathematics
- 2013

We consider the following measure of symmetry of a convex n-dimensional body K: $\rho(K)$ is the smallest constant for which there is a point x in K such that for partitions of K by an… Expand