5

Part of 2024 IMC

Problems(1)

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

Source: IMC 2024, Problem 5

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

f(p)

the probability that the convex hull of

x_1,\dots,x_n

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)

combinatorial geometryprobability