The Davenport constant of balls and boxes

Given an additively written abelian group $G$ and a set $X\subseteq G$, we let $\mathsf{D}(X)$ denote the Davenport constant of $X$, namely the largest non-negative integer $n$ for which there exists a sequence $x_1, \dots, x_n$ of elements of $X$ such that $\sum_{i=1}^n x_i =0$ and $\sum_{i \in I} x_i \ne 0$ for each non-empty proper subset $I$ of $\{1, \ldots, n\}$. In this paper, we mainly investigate the case when $G$ is $\mathbb{Z}^2$ and $\mathbb{Z}^3$, and $X$ is a discrete Euclidean ball. An application to the classical problem of estimating the Davenport constant of a box - a product of intervals of integers - is then obtained.