Let X, X1,..., Xn be independent and uniformly distributed on [0, 1]d. Prove                              E  min i=1,...,n X − Xi  ≥ d d + 1  Γ  d XXXXXXXXXX/d √π  · 1 n1/d . Hint: The volume of a...

Let X, X1,..., Xn be independent and uniformly distributed on [0, 1]d. Prove

                             E  min i=1,...,n X − Xi  ≥ d d + 1  Γ  d 2 + 1 1/d √π  · 1 n1/d .

Hint: The volume of a ball in Rd with radius t is given by

                     πd/2 Γ  d 2 + 1 · t d,

where Γ(x) = ‑ ∞ 0 t x−1e−t dt (x > 0) satisfies Γ(x + 1) = x · Γ(x), Γ(1) = 1, and Γ(1/2) = √π. Show that this implies

                  P  min i=1,...,n X − Xi ≤ t  ≤ n · πd/2 Γ d 2 + 1 · t d.