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I am trying to read about random polytopes, the convex hull of n random points $x_1,\ldots, x_n$ chosen independently inside a convex body $K$ with respect to uniformly distribution. My questions are:

  1. Does "uniformly distribution" mean that if we choose $x_1,...,x_n$ in a convex body $K$, then for any $Q\subset K$, we have $\text{Prob}(x_i\in Q)=\dfrac{V(Q)}{V(K)}$? Here $V$ denotes volume and $\text{Prob}$ means probability.

  2. With this distribution, why "with probability one, the random polytope is simplicial"?

  3. Is there any other distribution considered to create random polytopes?

I would be grateful for any hint on references or literature.

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For 1, any (Lebesgue) measurable $Q$ will do.

In 2, note that the probability that $n+2$ points lie in an affine $n$-subspace is zero.

For 3, one could choose random hyperplanes to serve as the polytope's facets.

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