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 $$\R^d$$. 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 $$\R^d$$.
 * The following parameterization has been used: $$r:(\R^d \times \{0,1\})^m\rightarrow \text{Polytopes} \in \R^d$$ such that $$r((p_1, 0), (p_2, 1), (p_3, 1)...(p_m, i_m)) = \{x \in \R^n: | \frac{p_j}{||p_j||} \cdot x \leq ||p_j|| \text{ if } i_j=1, \frac{p_j}{||p_j||} \cdot x \geq ||p_j|| \text{ if } i_j=0 \}$$ (Note: these polytopes can be empty).

Properties definition 1
Let $$\Kappa $$ be the set of convex bodies in $$\R^d$$. Assume $$K \in\Kappa $$ and consider a set of uniformly distributed points $$x_1, ..., x_n$$ in $$K $$. The convex hull of these points, $$K_n $$, is called a random polytope inscribed in $$K $$. $$K_n = [x_1, ..., x_n] $$ where the set $$[S] $$ stands for the convex hull of the set. We define $$E(k,n)$$ to be the expected volume of $$K - K_n$$. For a large enough $$n$$ and given $$K \in \R^n$$.


 * vol $$K(\frac{1}{n})\ll E(K,n) \ll$$ vol $$K(\frac{1}{n})$$
 * Note: One can determine the volume of the wet part to obtain the order of the magnitude of $$E(K,n) $$, instead of determining $$E(K,n)$$.
 * For the unit ball $$B^d \in \R^d$$, the wet part $$B^d(v \leq t)$$ is the annulus $$\frac{B^d}{(1-h)B^d}$$ where h is of order $$t^{\frac{2}{d+1}}$$: $$E(B^d,n) \approx $$ vol $$B^d (\frac{1}{n}) \approx n^{\frac{-2}{d+1}}$$

Given we have $$V = V(x_1,...,x_d)$$ is the volume of a smaller cap cut off from $$K$$ by aff$$(x_1,...,x_d)$$, and $$F=[x_1,...,x_d]$$ is a facet if and only if $$x_{d+1},...,x_n$$ are all on one side of aff $$\{x_1,...,x_d\}$$.


 * $$E_{\phi}(K_n) = {{n}\choose{d}} \int_K ... \int_K [(1-V)^{n-d} + V^{n-d}]\phi(F)dx_1...dx_d$$.
 * Note: If $$\phi = f_{d-1}$$ (a function that returns the amount of d-1 dimensional faces), then $$\phi(F) = 1$$ 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 $$(\R^d \times \{ 0, 1\})^m=(p_1\times i_1,\dots,p_m\times i_m)^m$$ that is


 * 1) Absolutely continuous on $$(p_1,\dots,p_d)$$ with respect to Lebesgue measure.
 * 2) Generates either 0 or 1 for the $$i$$s with probability of $$\frac{1}{2}$$ each.
 * 3) Assigns a measure of 0 to the set of elements in $$(\R^d \times \{ 0, 1\})^m$$ that correspond to empty polytopes.

Given this distribution, and our assumptions, the following properties hold:


 * A formula is derived for the expected number of $$k$$ dimensional faces on a polytope in $$\R^d$$ with $$m$$ constraints: $$E_k(m) = 2^{d-k} \sum_{i = d - k}^{d}{{i}\choose{d-k}}{{m}\choose{i}}/\sum_{i=0}^{d}{{m}\choose{i}}$$. (Note: $$\lim_{m \to \infty}E_k(m) = {{d}\choose{d-k}}2^{d-k}$$ where $$m>d$$). The upper bound, or worst case, for the number of vertices with $$m$$ constraints is much larger: $$V_{max} = {m-[\frac{1}{2}(d+1)]\choose m-d} + {m-[\frac{1}{2}(d+2)]\choose m-d}$$.
 * The probability that a new constraint is redundant is: $$\pi_{m} = 1 - \frac{2\sum_{i=0}^{d-1}{{m-1}\choose i}}{\sum_{i=0}^{d}{m\choose i}}$$. (Note: $$\lim_{m \to \infty}{\pi_m} = 1$$, and as we add more constraints, the probability a new constraint is redundant approaches 100%).
 * The expected number of non-redundant constraints is: $$C_d(m) = \frac{2m\sum_{i=0}^{d-1}{{m-1}\choose i}}{\sum_{i=0}^{d}{{m}\choose i}}$$. (Note: $$\lim_{m \to \infty}C_d(m) = 2d$$).

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)