This is a question from Brank Grunbaum's "Convex Polytopes":
Let $\{K_v\} $ be a family of convex sets in $\mathbb R^d $ such that every denumerable subfamily has non empty intersection, then $\cap_v K_v \neq \emptyset$
The issue here(and probably the whole point) is that the family of sets is not countable and therefore any sort of argument like take a point in one intersection, then replace one set with another get another point and so on and then from these points try to come up with a point in every set wouldn't work(something like helly's theorem).
Another idea I had was to use the fact that these sets are in $\mathbb R^d$ and conclude sort of a "at most $d$ linearly independent vectors" for how many convex sets there can be such that non of them is the other's convex hull (and WLOG assuming every convex set in the family is a convex hull of some set of points) but couldn't come up with anything of that sort.
Any ideas?
