  Denote by ${\mathfrak s}({\mathbb F}_p^d)$ the Erd{\H o}s--Ginzburg--Ziv
constant of ${\mathbb F}_p^d$, that is, the minimum $s$ such that any sequence
of $s$ vectors in ${\mathbb F}_p^d$ contains $p$ vectors whose sum is zero. Let
${\mathfrak w}({\mathbb F}_p^d)$ be the maximum size of a sequence of vectors
$v_1, \ldots, v_s \in {\mathbb F}_p^d$ such that for any integers $\alpha_1,
\ldots, \alpha_s \ge 0$ with sum $p$ we have $\alpha_1 v_1 + \ldots + \alpha_s
v_s \neq 0$ unless $\alpha_i = p$ for some $i$.
  In 1995, Alon--Dubiner proved that ${\mathfrak s}(\mathbb F_p^d)$ grows
linearly in $p$ when $d$ is fixed. In this work, we determine the constant of
linearity: for fixed $d$ and growing $p$ we show that ${\mathfrak s}({\mathbb
F}_p^d) \sim {\mathfrak w}({\mathbb F}_p^d) p$. Furthermore, for any $p$ and
$d$ we show that ${\mathfrak w}({\mathbb F}_p^d) \le {2d-1 \choose d}+1$. In
particular, ${\mathfrak s}({\mathbb F}_p^d) \le 4^d p$ for all sufficiently
large $p$ and fixed $d$.
