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How likely is it to be contained in the convex hull of n random points?

Source: IMC 2024, Problem 5

August 7, 2024

combinatorial geometryprobability

Problem Statement

Let

n>d

be positive integers. Choose

n

independent, uniformly distributed random points

x_1,\dots,x_n

in the unit ball

B \subset \mathbb{R}^d

centered at the origin. For a point

p \in B

denote by

f(p)

the probability that the convex hull of

x_1,\dots,x_n

contains

p

. Prove that if

p,q \in B

and the distance of

p

from the origin is smaller than the distance of

q

from the origin, then

f(p) \ge f(q)

.

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