Random polytope
In mathematics, a random polytope is a structure commonly used in convex analysis  and the analysis of linear programs in d-dimensional Euclidean space . Depending on use the construction and definition, random polytopes may differ. Definition 

There are multiple non equivalent definitions of a Random polytope. For the following definitions. Let K be a bounded convex set in a Euclidean space:
 The convex hull of random points selected with respect to a uniform distribution inside K.
 The nonempty intersection of half-spaces in . The following parameterization has been used:  such that  (Note: these polytopes can be empty). Properties definition 1  
Let  be the set of convex bodies in . Assume  and consider a set of uniformly distributed points  in . The convex hull of these points, , is called a random polytope inscribed in . where the set  stands for the convex hull of the set. We define  to be the expected volume of . For a large enough  and given . vol  vol 
 Note: One can determine the volume of the wet part to obtain the order of the magnitude of , instead of determining . For the unit ball , the wet part  is the annulus  where h is of order :  vol 

Given we have  is the volume of a smaller cap cut off from  by aff, and  is a facet if and only if  are all on one side of aff . . Note: If  (a function that returns the amount of d-1 dimensional faces), then  and formula can be evaluated for smooth convex sets and for polygons in the plane. Properties definition 2  
Assume we are given a multivariate probability distribution on  that is

 Absolutely continuous on  with respect to Lebesgue measure. Generates either 0 or 1 for the s with probability of  each. Assigns a measure of 0 to the set of elements in  that correspond to empty polytopes. Given this distribution, and our assumptions, the following properties hold:

 A formula is derived for the expected number of  dimensional faces on a polytope in  with  constraints: . (Note:  where ). The upper bound, or worst case, for the number of vertices with  constraints is much larger: . The probability that a new constraint is redundant is: . (Note: , and as we add more constraints, the probability a new constraint is redundant approaches 100%). The expected number of non-redundant constraints is: . (Note: ). Example uses 

 Minimal caps
 Macbeath regions
 Approximations (approximations of convex bodies see properties of definition 1)
 Economic cap covering theorem (see relation from properties of definition 1 to floating bodies)

References 

Metric geometry
Convex analysis
Computational geometry