# Strictly convex set

When a function is strictly convex it has many desirable properties, most notably that it admits a unique minimum.

I was wondering if there is anything desirable about a strictly convex set (meaning that for any two points, all points between lie strictly within the set)? In the problems I've seen, most only care to distinguish between the convexity and non-convexity of a set and leave it at that. Does anyone know of any applications where strict convexity of a set is taken advantage of?

• I'm late to the party here but $f(x)\equiv 0$ will not have a unique minimum on any set with more than one point. – Michael Grant Jan 6 '17 at 12:47