English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Zendy Plus

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Zendy Tools

Zendy Open

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