Question

Question: Find an example of a closed convex set S in \({\mathbb{R}^{\bf{2}}}\) such that its profile P is nonempty but \({\bf{conv}}\,P \ne S\).

Short answer

A rectangle without lower and right boundaries represents a bounded convex set S in \({\mathbb{R}^2}\) such that profile P is nonempty but \({\rm{conv}}\,\,P \ne S\).

Step-by-step solution

Find the example of a closed convex set

One example of such a closed convex set is a rectangle that has all parts except lower and right boundary as shown below:

Create the convex hull

The convex hull can be created by outlining a triangular region with three of its vertices.

It represents one example of a bounded convexset S in \({\mathbb{R}^2}\) such that its profile P in non empty but \({\rm{conv}}\,\,P \ne S\).

So, a rectangle without lower and right boundary represents a bounded convex set S in \({\mathbb{R}^2}\) such that profile P is nonempty but \({\rm{conv}}\,\,P \ne S\).