Threshold for the expected measure of random polytopes

Let $\mu$ be a log-concave probability measure on ${\mathbb{R}}^n$ and for any $N>n$ consider the random polytope $K_N=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{v}\{X_1,\dots ,X_N\}$, where $X_1,X_2,\dots$ are independent random points in ${\mathbb{R}}^n$ distributed according to $\mu$. We study the question if there exists a threshold for the expected measure of $K_N$. Our approach is based on the Cramer transform ${\mathrm{\Lambda }}_{\mu }^{\ast }$ of $\mu$. We examine the existence of moments of all orders for ${\mathrm{\Lambda }}_{\mu }^{\ast }$ and establish, under some conditions, a sharp threshold for the expectation ${\mathbb{E}}_{{\mu }^N}[\mu (K_N)]$ of the measure of $K_N$: it is close to $0$ if $\ln N\ll {\mathbb{E}}_{\mu }({\mathrm{\Lambda }}_{\mu }^{\ast })$ and close to $1$ if $\ln N\gg {\mathbb{E}}_{\mu }({\mathrm{\Lambda }}_{\mu }^{\ast })$. The main condition is that the parameter $\beta (\mu )={\mathrm{V}\mathrm{a}\mathrm{r}}_{\mu }({\mathrm{\Lambda }}_{\mu }^{\ast })\mathrm{/}({\mathbb{E}}_{\mu }({\mathrm{\Lambda }}_{\mu }^{\ast }){)}^2$ should be small.