  We examine the averaging operator $T$ corresponding to the manifold in
$\mathbb{R}^{2d}$ of pairs of points $(u,v)$ satisfying $|u| = |v| = |u - v| =
1$, so that $\{0,u,v\}$ is the set of vertices of an equilateral triangle. We
establish $L^p \times L^q \rightarrow L^r$ boundedness for $T$ for $(1/p, 1/q,
1/r)$ in the convex hull of the set of points $\lbrace (0, 0, 0) ,\, (1, 0 , 1)
,\, (0, 1, 1) , \, ({1}/{p_d}, {1}/{p_d}, {2}/{p_d}) \rbrace$, where $p_d =
\frac{19d-4}{11d - 12}$ and $d\geq 7$.
