Convex hull $H$, of set of points $P=\{p_i: p_i \in R^2\}$, always exists if $|P|\ge3$. I wanted to know if the convex hull of an empty set, one-point set, or a two-point set is the set itself?
i.e., $H(\{\})$, $H(\{p_1\})$, $H(\{p_1,p_2\})$
How to prove this if it does?
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rohitt
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Do you have a definition of "convex hull of set $S\subseteq\mathbb R^n$"? If so then have a good look at it and decide for yourself. – drhab Jan 06 '22 at 14:19
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It's true for a one set point. The set is convex, and it is the "smallest" convex set containing the point, so it is the convex hull.
For a two set point, the convex hull is simply the line segment connecting the 2 points. Remember the definition of a convex set: With 2 points in the set, any point on their connecting line segment has to be in it as well. So the connecting line segment has to be in the convex hull, and since the line segment is convex, it is the convex hull.
The empty set itself is convex, so the convex hull of it is the empty set itself again.
Ingix
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If one says the empty set is convex, then the arbitrary intersection of convex sets is convex. If you say the empty set is not convex, then the intersection of two disjoint convex sets is not convex. – Jay Jan 06 '22 at 14:41
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Yes, that's true. But I never came across the need to consider the convex hull of the empty set. That's what my "you can say" refers to, but I see now that it is worded open to other interpretations. Will reword. – Ingix Jan 06 '22 at 15:02