Dimension-raising Homomorphisms between Lattices of Convex Bodies

TL;DR

This paper fully characterizes dimension-raising homomorphisms between lattices of convex bodies in Euclidean spaces when the dimension difference is one, resolving a long-standing open problem and applying results to convex functions.

Contribution

It provides a complete description of homomorphisms between convex body lattices for the case when the target space dimension is exactly one higher, solving a problem posed in 1991.

Findings

Characterization of homomorphisms for d=c+1, c ≥ 3

Application to anti-homomorphisms and homomorphisms to convex functions

Resolution of a problem posed by P. M.. Gruber in 1991

Abstract

We settle the first unsolved case of a problem of P. M. Gruber, asked by him in 1991, namely, to investigate the homomorphisms from the lattice of convex bodies of ${\mathbb{E}}^c$ to the lattice of convex bodies of ${\mathbb{E}}^d$ for $c<d$. We completely describe these homomorphisms for the case $d=c+1$, for $c \ge 3$. The obtained result is then applied to characterize anti-homomorphisms and homomorphisms from lattices of convex bodies to lattices of convex functions.