Convex bodies and graded families of monomial ideals

TL;DR

This paper establishes a deep connection between convex geometry and commutative algebra by showing that mixed volumes of convex bodies correspond to mixed multiplicities of graded families of monomial ideals, linking two mathematical theories.

Contribution

It proves that mixed volumes of convex bodies are equal to mixed multiplicities of graded families of monomial ideals, revealing a fundamental relationship between convex geometry and algebra.

Findings

Mixed volumes equal mixed multiplicities of monomial ideals

Normalized limits of mixed multiplicities match mixed volumes

Highlights the connection between convex geometry and algebra

Abstract

We show that the mixed volumes of arbitrary convex bodies are equal to mixed multiplicities of graded families of monomial ideals, and to normalized limits of mixed multiplicities of monomial ideals. This result evinces the close relation between the theories of mixed volumes from convex geometry and mixed multiplicities from commutative algebra.