A metric on the moduli space of bodies

TL;DR

This paper introduces a new metric on the moduli space of bodies in Euclidean space, connecting geometric and symplectic structures through affine transformations and Delzant polytopes.

Contribution

It constructs a metric on the moduli space of bodies, linking it to symplectic toric manifolds via Delzant polytopes, expanding geometric understanding.

Findings

Defined a metric on the moduli space of bodies

Identified a subspace corresponding to symplectic toric manifolds

Discussed related geometric problems

Abstract

We construct a metric on the moduli space of bodies in Euclidean space. The moduli space is defined as the quotient space with respect to the action of integral affine transformations. This moduli space contains a subspace, the moduli space of Delzant polytopes, which can be identified with the moduli space of symplectic toric manifolds. We also discuss related problems.