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Difference between revisions of "Convex hull"

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''of a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026270/c0262701.png" />''
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''of a set $M$''
  
The minimal [[Convex set|convex set]] containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026270/c0262702.png" />; it is the intersection of all convex sets containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026270/c0262703.png" />. The convex hull of a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026270/c0262704.png" /> is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026270/c0262705.png" />. In the Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026270/c0262706.png" /> the convex hull is the set of possible locations of the centre of gravity of a mass which can be distributed in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026270/c0262707.png" /> in different manners. Each point of the convex hull is the centre of gravity of a mass concentrated at not more than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026270/c0262708.png" /> points (Carathéodory's theorem).
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The minimal [[Convex set|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026270/c0262709.png" /> or is identical with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026270/c02627010.png" />. The part of the boundary of the convex hull not adjacent to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026270/c02627011.png" /> has the local structure of a developable hypersurface. In <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026270/c02627012.png" /> the convex hull of a bounded closed set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026270/c02627013.png" /> is the convex hull of the extreme points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026270/c02627014.png" /> (an extreme point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026270/c02627015.png" /> is a point of this set which is not an interior point of any segment belonging to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026270/c02627016.png" />).
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026270/c02627017.png" />. In <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026270/c02627018.png" />, the convex hull of a compact set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026270/c02627019.png" /> is the closed convex hull of its extreme points (the Krein–Mil'man theorem).
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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====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  R.E. Edwards,  "Functional analysis: theory and applications" , Holt, Rinehart &amp; Winston  (1965)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  R.R. Phelps,  "Lectures on Choquet's theorem" , v. Nostrand  (1966)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  R.E. Edwards,  "Functional analysis: theory and applications" , Holt, Rinehart &amp; Winston  (1965)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  R.R. Phelps,  "Lectures on Choquet's theorem" , v. Nostrand  (1966)</TD></TR></table>

Revision as of 14:28, 10 April 2014

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=31501
This article was adapted from an original article by V.A. Zalgaller (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article