  The ball hypergraph of a graph $G$ is the family of balls of all possible
centers and radii in $G$. It has Helly number at most $k$ if every subfamily of
$k$-wise intersecting balls has a nonempty common intersection. A graph is
$k$-Helly (or Helly, if $k=2$) if its ball hypergraph has Helly number at most
$k$. We prove that a central vertex and all the medians in an $n$-vertex
$m$-edge Helly graph can be computed w.h.p. in $\tilde{\cal O}(m\sqrt{n})$
time. Both results extend to a broader setting where we define a non-negative
cost function over the vertex-set. For any fixed $k$, we also present an
$\tilde{\cal O}(m\sqrt{kn})$-time randomized algorithm for radius computation
within $k$-Helly graphs. If we relax the definition of Helly number (for what
is sometimes called an "almost Helly-type" property in the literature), then
our approach leads to an approximation algorithm for computing the radius with
an additive one-sided error of at most some constant.
