Threshold for the expected measure of random polytopes

Abstract

AbstractLet $$\mu $$ μ be a log-concave probability measure on $${\mathbb R}^n$$ R n and for any $$N>n$$ N > n consider the random polytope $$K_N=\textrm{conv}\{X_1,\ldots ,X_N\}$$ K N = conv { X 1 , … , X N } , where $$X_1,X_2,\ldots $$ X 1 , X 2 , … are independent random points in $${\mathbb R}^n$$ R n distributed according to $$\mu $$ μ . We study the question if there exists a threshold for the expected measure of $$K_N$$ K N . Our approach is based on the Cramer transform $$\Lambda _{\mu }^{*}$$ Λ μ ∗ of $$\mu $$ μ . We examine the existence of moments of all orders for $$\Lambda _{\mu }^{*}$$ Λ μ ∗ and establish, under some conditions, a sharp threshold for the expectation $${\mathbb {E}}_{\mu ^N}[\mu (K_N)]$$ E μ N [ μ ( K N ) ] of the measure of $$K_N$$ K N : it is close to 0 if $$\ln N\ll {\mathbb {E}}_{\mu }(\Lambda _{\mu }^{*})$$ ln N ≪ E μ ( Λ μ ∗ ) and close to 1 if $$\ln N\gg {\mathbb {E}}_{\mu }(\Lambda _{\mu }^{*})$$ ln N ≫ E μ ( Λ μ ∗ ) . The main condition is that the parameter $$\beta (\mu )=\textrm{Var}_{\mu }(\Lambda _{\mu }^{*})/({\mathbb {E}}_{\mu }(\Lambda _{\mu }^{*}))^2$$ β ( μ ) = Var μ ( Λ μ ∗ ) / ( E μ ( Λ μ ∗ ) ) 2 should be small.