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Fix a space dimension $dge 2$, parameters $alpha > -1$ and $beta ge 1$, and let $gamma_{d,alpha, beta}$ be the probability measure of an isotropic random vector in $mathbb{R}^d$ with density proportional to begin{align*} ||x||^alpha, expleft(-frac{|x|^beta}{beta}right), qquad xin mathbb{R}^d. end{align*} By $K_lambda$, we denote the Generalized Gamma Polytope arising as the random convex hull of a Poisson point process in $mathbb{R}^d$ with intensity measure $lambdagamma_{d,alpha,beta}$, $lambda>0$. We establish that the scaling limit of the boundary of $K_lambda$, as $lambda rightarrow infty$, is given by a universal `festoon of piecewise parabolic surfaces, independent of $alpha$ and $beta$. Moreover, we state a list of other large scale asymptotic results, including expectation and variance asymptotics, central limit theorems, concentration inequalities, Marcinkiewicz-Zygmund-type strong laws of large numbers, as well as moderate deviation principles for the intrinsic volumes and face numbers of $K_lambda$.
Let $K_n$ be the convex hull of i.i.d. random variables distributed according to the standard normal distribution on $R^d$. We establish variance asymptotics as $n to infty$ for the re-scaled intrinsic volumes and $k$-face functionals of $K_n$, $k in
Let K be a convex set in R d and let K $lambda$ be the convex hull of a homogeneous Poisson point process P $lambda$ of intensity $lambda$ on K. When K is a simple polytope, we establish scaling limits as $lambda$ $rightarrow$ $infty$ for the boundar
The convex hull generated by the restriction to the unit ball of a stationary Poisson point process in the $d$-dimensional Euclidean space is considered. By establishing sharp bounds on cumulants, exponential estimates for large deviation probabiliti
We consider the eigenvalues of symplectic elliptic Ginibre matrices which are known to form a Pfaffian point process whose correlation kernel can be expressed in terms of the skew-orthogonal Hermite polynomials. We derive the scaling limits and the c
Suppose we choose $N$ points uniformly randomly from a convex body in $d$ dimensions. How large must $N$ be, asymptotically with respect to $d$, so that the convex hull of the points is nearly as large as the convex body itself? It was shown by Dyer-