Intersection of convex sets

[francesco]

Given n regular bounded convex sets in R^{k} such that the intersection of all boundaries is non empty. Take the ceiling part of n/2 +1=M.
if n=5 then M=3, n=7 then M=4 and so on.
And n is an odd integer. The convex sets are k dimensional subsets in the Lebesque measure's sense.

Can I say that M interiors of the initial convex sets have an intersection non empty?

P.S. The numbers n and k are not dependent.

Thanks for your help.