# 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 Andreas 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…

## 7 Citations

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## References

SHOWING 1-10 OF 22 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.…

### Greatest lower bounds on the Ricci curvature of Fano manifolds

- MathematicsCompositio 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…

### 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…

### 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…

### Complete classification of reflexive polyhedra in four dimensions

- 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…

### 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…

### Introduction to toric varieties

- Mathematics
- 2004

The course given during the School and Workshop “The Geometry and Topology of Singularities”, 8-26 January 2007, Cuernavaca, Mexico is based on a previous course given during the 23o Coloquio…

### Entropy and Asymptotic Geometry of Non-Symmetric Convex Bodies☆

- Mathematics
- 2000

Abstract We extend to the general, not necessarily centrally symmetric setting a number of basic results of local theory which were known before for centrally symmetric bodies and were using very…