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 $\vec{X}_1,\ldots ,\vec{X}_N$ 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 $K_N:={\rm conv}\bigl\{\vec{X}_1,\ldots,\vec{X}_N\bigr\}$ in ${\mathbb R}^n$ under the assumption that the Legendre transform $\Lambda_{\mu}^{\ast}$ of the logarithmic moment generating function of $\mu$ satisfies the condition $$\lim\limits_{x\uparrow x^{\ast}}\dfrac{-\ln \mu ([x,\infty ))}{\Lambda_{\mu}^{\ast}(x)}=1,$$ where $x^{\ast}=\sup\{x\in\mathbb{R}\colon \mu([x,\infty))>0\}$. An application is a sharp threshold for the case of the product measure $\nu_p^n=\nu_p^{\otimes n}$, $p\geq 1$ with density $(2\gamma_p)^{-n}\exp(-\|x\|_p^p)$, where $\|\cdot\|_p$ is the $\ell_p^n$-norm and $\gamma_p=\Gamma(1+1/p)$.