Convexity/Convex polytopes

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Definition: A convex polytope is the convex hull of a finite number of points. Usually, there will be at least three non-collinear points.

Theorem: A set is a convex polytope if and only if:

  1. It is not the empty set
  2. It is bounded
  3. It is the intersection of a finite number of closed half-spaces.

A simplex in an n-dimensional vector space is the convex hull of n+1 points that do not all lie on the same hyperplane. If n=2, a simplex is a triangle; if n=3, it is a tetrahedron.