  Assume that $k \le d$ is a positive integer and $\C$ is a finite collection
of convex bodies in $\R^d$. We prove a Helly type theorem: If for every
subfamily $\C^*\subset \C$ of size at most $\max \{d+1,2(d-k+1)\}$ the set
$\bigcap \C^*$ contains a $k$-dimensional cone, then so does $\bigcap \C.$ One
ingredient in the proof is another Helly type theorem about the dimension of
lineality spaces of convex cones.
