Non-zero probability of hitting a convex hull of d + 1 i.i.d. points in R...

iarc6io

iarc6io

Answered

2022-07-18

Non-zero probability of hitting a convex hull of d + 1 i.i.d. points in R d .
Let X 1 , , X d + 1 be d + 1 i.i.d. random points in R d sampled from a continuous probability μ of density f.
Let x 0 R d . Is it true that almost surely with respect to μ, P ( x 0 C o n v ( X 1 , , X d + 1 ) ) > 0 where Conv denotes the convex hull of the points?

Answer & Explanation

Alanna Downs

Alanna Downs

Expert

2022-07-19Added 11 answers

Step 1
Choose ( d + 1 ) balls B 1 , , B d + 1 in R d such that 0 Conv ( x 1 , , x d + 1 ) for any choices of x i B i for i = 1 , , d + 1. For instance, fix a regular d-simplex centered at 0 and replace each vertex with a small ball.
Step 2
Now let x 0 be a Lebesgue point of f such that f ( x 0 ) > 0. Then it is straightforward to prove that
lim r 0 + 1 r d | B i | x 0 + r B i f ( x ) d x = f ( x 0 ) > 0
for each i = 1 , , d + 1. So if r > 0 is sufficiently small, then
P ( x 0 Conv ( X 1 , , X d + 1 ) ) P ( i = 1 d + 1 { X i x 0 + r B i } ) > 0.

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