Threshold for the expected measure of the convex hull of random points with independent coordinates

Let $\mu$ be an even Borel probability measure on $\mathbb{R}$. For every $N>n$ consider $N$ independent random vectors ${\vec{X}}_1,\dots ,{\vec{X}}_N$ in ${\mathbb{R}}^n$, with independent coordinates having distribution $\mu$. We establish a sharp threshold for the product measure ${\mu }_n$ of the random polytope $K_N:=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{v}\{{\vec{X}}_1,\dots ,{\vec{X}}_N\}$ in ${\mathbb{R}}^n$ under the assumption that the Legendre transform ${\mathrm{\Lambda }}_{\mu }^{\ast }$ of the logarithmic moment generating function of $\mu$ satisfies the condition $$\underset{x\uparrow x^{\ast }}{\lim }{\displaystyle \frac{-\ln \mu ([x,\mathrm{\infty }))}{{\mathrm{\Lambda }}_{\mu }^{\ast }(x)}}=1,$$ where $x^{\ast }=\sup \{x\in \mathbb{R}\text{\hspace{0.17em}⁣}:\mu ([x,\mathrm{\infty }))>0\}$. An application is a sharp threshold for the case of the product measure ${\nu }_p^n={\nu }_p^{\otimes n}$, $p\ge 1$ with density $(2{\gamma }_p{)}^{-n}\exp (-\mathrm{\parallel }x{\mathrm{\parallel }}_p^p)$, where $\mathrm{\parallel }\cdot {\mathrm{\parallel }}_p$ is the ${\mathrm{\ell }}_p^n$-norm and ${\gamma }_p=\mathrm{\Gamma }(1+1\mathrm{/}p)$.