  For $r \ge 2$, an $r$-uniform hypergraph is called a friendship
$r$-hypergraph if every set $R$ of $r$ vertices has a unique 'friend' - that
is, there exists a unique vertex $x \notin R$ with the property that for each
subset $A \subseteq R$ of size $r-1$, the set $A \cup \{x\}$ is a hyperedge.
  We show that for $r \geq 3$, the number of hyperedges in a friendship
$r$-hypergraph is at least $\frac{r+1}{r} \binom{n-1}{r-1}$, and we
characterise those hypergraphs which achieve this bound. This generalises a
result given by Li and van Rees in the case when $r = 3$.
  We also obtain a new upper bound on the number of hyperedges in a friendship
$r$-hypergraph, which improves on a known bound given by Li, van Rees, Seo and
Singhi when $r=3$.
