Distance from a point to a convex set

Question

Consider a sphere, which is obviously a convex set. Consider any point outside the sphere, and say I want to find the minimum distance to this set.

In this case I can intuitively see that the closest point will lie on the line between the centre of the sphere and the outside point. So it will be a convex combination of the centre C and the outside point K. $$pC + (1-p)K$$

Is this true for a point and a convex set in any dimension? That the point in the set closest to the outside point will be a convex combination of the centre and the outside point? If so, how would I prove this?

Would it also be true for any distance measure?

Answer 1

Umm..... why would you ever think that?