# 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://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