Question 10: Find an example of a closed convex set \(S\) in \({\mathbb{R}^2}\) such that its profile P is nonempty but \({\mathop{\rm conv}\nolimits} P \ne S\).

Let \(S\) be a convex set. A point p is called anextreme point of \(S\), if p is not in the interior of any line segment that lies in \(S\).

More precisely, when \({\mathop{\rm x}\nolimits} ,y \in S\) and \({\mathop{\rm p}\nolimits} \in \overline {{\mathop{\rm xy}\nolimits} } \), then \({\mathop{\rm p}\nolimits} = {\mathop{\rm x}\nolimits} \) or \({\mathop{\rm p}\nolimits} = {\mathop{\rm y}\nolimits} \). The set of all extreme points of \(S\)is referred asprofile of \(S\).

Consider \(S\) as a ray in \({\mathbb{R}^2}\). The profile \({\mathop{\rm P}\nolimits} \) contains the endpoint. The ray originates from \({\mathop{\rm P}\nolimits} \)so it is not empty.

But \({\mathop{\rm conv}\nolimits} P = P\) because \({\mathop{\rm P}\nolimits} \) contains only a single endpoint which is not the same as \(S\) and hence \({\mathop{\rm conv}\nolimits} P \ne S\).

Thus, \({\mathop{\rm P}\nolimits} \)is not empty but \({\mathop{\rm conv}\nolimits} P \ne S\).