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Let    μ  $\mu$    be an even Borel probability measure on    R  ${\mathbb {R}}$   . For everyN  &gt;  n   $N&amp;gt;n$   , consider    N  $N$    independent random vectorsX  ⃗   1   ,  …  ,    X  ⃗   N    $\vec{X}_1,\ldots ,\vec{X}_N$    inR  n   ${\mathbb {R}}^n$   , with independent coordinates having distribution    μ  $\mu$   . We establish a sharp threshold for the product measureμ  n   $\mu _n$    of the random polytopeK  N   :  =  conv   {    X  ⃗   1   ,  …  ,    X  ⃗   N   }    $K_N:=\operatorname{conv}\lbrace \vec{X}_1,\ldots ,\vec{X}_N\rbrace$    inR  n   ${\mathbb {R}}^n$    under the assumption that the Legendre transformΛ  μ  *   $\Lambda _{\mu }^{\ast }$    of the logarithmic moment generating function of    μ  $\mu$    satisfies the conditionlim    x  ↑    x  *−  ln  μ  (  [  x  ,  ∞  )  )Λ  μ  *    (  x  )=  1  ,   $$\begin{equation*} \hspace*{6pc}\lim \limits _{x\uparrow x^{\ast }}\dfrac{-\ln \mu ([x,\infty ))}{\Lambda _{\mu }^{\ast }(x)}=1,\hspace*{-6pc} \end{equation*}$$    wherex  *   =  sup   {  x  ∈  R  :  μ   (   [  x  ,  ∞  )   )   &gt;  0  }    $x^{\ast }=\sup \lbrace x\in \mathbb {R}\colon \mu ([x,\infty ))&amp;gt;0\rbrace$   . An application is a sharp threshold for the case of the product measureν  p  n   =   ν  p   ⊗  n$\nu _p^n=\nu _p^{\otimes n}$   ,p  ⩾  1   $p\geqslant 1$    with density(  2   γ  p   )    −  nexp  (  −  ∥  x  ∥   p  p    )    $(2\gamma _p)^{-n}\exp (-\Vert x\Vert _p^p)$   , where∥  ·  ∥   p   $\Vert \cdot \Vert _p$    is theℓ  p  n   $\ell _p^n$   ‐norm andγ  p   =  Γ   (  1  +  1  /  p  )    $\gamma _p=\Gamma (1+1/p)$   .

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