A convex hull of a set $X$ of points is defined as:
the smallest convex set that contains $X$.
Does this definition imply that all convex hulls of a convex set are in fact, polytopes of their respective dimensions?
2026-03-25 20:35:43.1774470943
Are all convex hulls polytopes too?
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Your question seems to me to indicate several confusions.
Here are some facts you should understand. They follow from the definition of "convex hull".
A set has just one convex hull, so you needn't ask about "all convex hulls".
If a set is convex then it is its convex hull (as @YvesDaoust comments).
The convex hull of a finite set will be a polytope. Its dimension may be lower than the dimension of the containing space - for an example think about the convex hull of two points in the plane (or even the convex hull of one point on the line).
Any polytope is the convex hull of the (finite) set of its vertices.
The convex hull of (the circumference of) a circle in the plane is the circle together with its interior. Clearly that's not a polytope.