Editing Convex hull (section)

===Equivalence of definitions===
[[File:3D_Convex_Hull.tiff|thumb|3D convex hull of 120 point cloud]]
It is not obvious that the first definition makes sense: why should there exist a unique minimal convex set containing <math>X</math>, for every <math>X</math>? However, the second definition, the intersection of all convex sets containing <math>X</math>, is well-defined. It is a subset of every other convex set <math>Y</math> that contains <math>X</math>, because <math>Y</math> is included among the sets being intersected. Thus, it is exactly the unique minimal convex set containing <math>X</math>. Therefore, the first two definitions are equivalent.{{sfnp|Rockafellar|1970|page=12}}

Each convex set containing <math>X</math> must (by the assumption that it is convex) contain all convex combinations of points in <math>X</math>, so the set of all convex combinations is contained in the intersection of all convex sets containing <math>X</math>. Conversely, the set of all convex combinations is itself a convex set containing <math>X</math>, so it also contains the intersection of all convex sets containing <math>X</math>, and therefore the second and third definitions are equivalent.<ref name=rock12lay17>{{harvtxt|Rockafellar|1970}}, p. 12; {{harvtxt|Lay|1982}}, p. 17.</ref>

In fact, according to [[Carathéodory's theorem (convex hull)|Carathéodory's theorem]], if <math>X</math> is a subset of a <math>d</math>-dimensional Euclidean space, every convex combination of finitely many points from <math>X</math> is also a convex combination of at most <math>d+1</math> points in <math>X</math>. The set of convex combinations of a <math>(d+1)</math>-tuple of points is a [[simplex]]; in the plane it is a [[triangle]] and in three-dimensional space it is a tetrahedron. Therefore, every convex combination of points of <math>X</math> belongs to a simplex whose vertices belong to <math>X</math>, and the third and fourth definitions are equivalent.<ref name=rock12lay17/>