Convex vs strict convex sets

Question

Could somebody give me an example of a strict convex set? I can't find any info on the internet more than the definition and I have a hard time getting an intuition for the difference between convex set and strict convex. So an example in $\Re$,$\Re^2$ or $\Re^3$ would be very appreciated. Thanks

Answer 1

A set $C \subseteq \mathbb{R}^n$ is convex if it contains all line segments between any two of its points. It is strictly convex if, furthermore, such a line segment does not intersect the boundary $\partial C$, except possibly at its endpoints.

For example, the unit $n$-ball $D^n \subseteq \mathbb{R}^n$ is strictly convex, but the unit $n$-cube $[0,1]^n \subseteq \mathbb{R}^n$ is not. (For the former, note that the line segment connecting any two vertices of the $n$-cube is contained entirely in the boundary of the $n$-cube.)