  Let $\mathbb{T}$ denote a rooted $b$-ary tree and let $\{S_v\}_{v\in
\mathbb{T}}$ denote a branching random walk indexed by the vertices of the
tree, where the increments are i.i.d. and possess a logarithmic moment
generating function $\Lambda(\cdot)$. Let $m_n$ denote the minimum of the
variables $S_v$ over all vertices at the $n$th generation, denoted by
$\mathbb{D}_n$. Under mild conditions, $m_n/n$ converges almost surely to a
constant, which for convenience may be taken to be 0. With $\bar
S_v=\max\{S_w:{\rm $w$ is on the geodesic connecting the root to $v$}\}$,
define $L_n=\min_{v\in \mathbb{D}_n} \bar S_v$. We prove that $L_n/n^{1/3}$
converges almost surely to an explicit constant $l_0$. This answers a question
of Hu and Shi.
