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Convex hull

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of a set $M$

The minimal convex set containing $M$; it is the intersection of all convex sets containing $M$. The convex hull of a set $M$ is denoted by $\operatorname{conv} M$. In the Euclidean space $E^n$ the convex hull is the set of possible locations of the centre of gravity of a mass which can be distributed in $M$ in different manners. Each point of the convex hull is the centre of gravity of a mass concentrated at not more than $n+1$ points (Carathéodory's theorem).

The closure of the convex hull is called the closed convex hull. It is the intersection of all closed half-spaces containing $M$ or is identical with $E^n$. The part of the boundary of the convex hull not adjacent to $M$ has the local structure of a developable hypersurface. In $E^n$ the convex hull of a bounded closed set $M$ is the convex hull of the extreme points of $M$ (an extreme point of $M$ is a point of this set which is not an interior point of any segment belonging to $M$).

In addition to Euclidean spaces, convex hulls are usually considered in locally convex linear topological spaces $L$. In $L$, the convex hull of a compact set $M$ is the closed convex hull of its extreme points (the Krein–Mil'man theorem).

References

[1] R.E. Edwards, "Functional analysis: theory and applications" , Holt, Rinehart & Winston (1965)
[2] R.R. Phelps, "Lectures on Choquet's theorem" , v. Nostrand (1966)
How to Cite This Entry:
Convex hull. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Convex_hull&oldid=34461
This article was adapted from an original article by V.A. Zalgaller (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article