# Smooth polytopes with negative Ehrhart coefficients

## TL;DR

This paper constructs smooth lattice polytopes in higher dimensions with negative Ehrhart coefficients, answering a longstanding question and highlighting the role of Berline-Vergne valuations in Ehrhart positivity.

## Contribution

It provides explicit examples of smooth polytopes with negative Ehrhart coefficients and discusses the application of Berline-Vergne valuations in Ehrhart theory.

## Key findings

- Existence of smooth polytopes with negative Ehrhart coefficients in dimensions 3 and higher
- Use of Berline-Vergne valuations to analyze Ehrhart positivity
- Confirmation that negative coefficients can occur in smooth lattice polytopes

## Abstract

We present examples of smooth lattice polytopes in dimensions 3 and higher where each coefficient of their Ehrhart polynomials that can potentially be negative is indeed negative. This answers a question by Bruns. We also discuss Berline-Vergne valuations as a useful tool in proving Ehrhart positivity results.