Reconstruction of symmetric convex bodies from Ehrhart-like data

TL;DR

This paper extends previous work on reconstructing polytopes from Ehrhart functions to symmetric convex bodies, demonstrating that Ehrhart-like data uniquely determines such bodies.

Contribution

The paper proves that symmetric convex bodies can be uniquely reconstructed from Ehrhart-like data, generalizing prior results from rational polytopes.

Findings

Unique reconstruction of symmetric convex bodies from Ehrhart data

Extension of Ehrhart-based reconstruction to broader class of bodies

Theoretical proof of equivalence for symmetric convex bodies

Abstract

In a previous paper, we showed how to use the Ehrhart function $L_P(s)$, defined by $L_P(s) = \#(sP \cap \mathbb Z^d)$, to reconstruct a polytope $P$. More specifically, we showed that, for rational polytopes $P$ and $Q$, if $L_{P + w}(s) = L_{Q + w}(s)$ for all integer vectors $w$, then $P = Q$. In this paper we show the same result, but assuming that $P$ and $Q$ are symmetric convex bodies instead of rational polytopes.