Editing Convex hull (section)

==Geometric and algebraic properties==
===Closure operator===
The convex-hull operator has the characteristic properties of a [[closure operator]]:{{sfnp|Kiselman|2002}}
*It is ''extensive'', meaning that the convex hull of every set <math>X</math> is a superset of <math>X</math>.
*It is ''[[Monotone function#Monotonicity in order theory|non-decreasing]]'', meaning that, for every two sets <math>X</math> and <math>Y</math> with <math>X\subseteq Y</math>, the convex hull of <math>X</math> is a subset of the convex hull of <math>Y</math>.
*It is ''[[idempotence|idempotent]]'', meaning that for every <math>X</math>, the convex hull of the convex hull of <math>X</math> is the same as the convex hull of <math>X</math>.
When applied to a finite set of points, this is the closure operator of an [[antimatroid]], the shelling antimatroid of the point set.
Every antimatroid can be represented in this way by convex hulls of points in a Euclidean space of high-enough dimension.{{sfnp|Kashiwabara|Nakamura|Okamoto|2005}}

===Minkowski sum===
The operations of constructing the convex hull and taking the [[Minkowski sum]] commute with each other, in the sense that the Minkowski sum of convex hulls of sets gives the same result as the convex hull of the Minkowski sum of the same sets. This provides a step towards the [[Shapley–Folkman lemma|Shapley–Folkman theorem]] bounding the distance of a Minkowski sum from its convex hull.<ref>{{harvtxt|Krein|Šmulian|1940}}, Theorem&nbsp;3, pages&nbsp;562–563; {{harvtxt|Schneider|1993}}, Theorem&nbsp;1.1.2 (pages&nbsp;2–3) and Chapter&nbsp;3.</ref>

===Projective duality===
The [[projective dual]] operation to constructing the convex hull of a set of points is constructing the intersection of a family of closed halfspaces that all contain the origin (or any other designated point).{{sfnp|de Berg|van Kreveld|Overmars|Schwarzkopf|2008|page=254}}