Positive bases, cones, Helly type theorems

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