Petty proved that a convex body in R^n has the minimal surface area amongst its SL(n)images, if, and only if, its surface area measure is isotropic. By introducing a new notion of minimal Orlicz surface area, we generalize this result to the Orlicz setting. The analog of Ball’s reverse isoperimetric inequality is established.

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For a random quantum state on $\cH=\C^d \otimes \C^d$ obtained by partial tracing a random pure state on $\cH \otimes \C^s$, we consider the question whether it is typically separable or typically entangled. For this problem, we show the existence of a sharp threshold $s_0=s_0(d)$ of order roughly $d^3$. More precisely, for any $\e>0$ and for $d$ large enough, such a random state is entangled with very large probability when $s \leq (1-\e)s_0$, and separable with very large probability when $s \geq (1+\e) s_0$. One consequence of this result is as follows: for a system of $N$ identical particles in a random pure state, there is a threshold $k_0 = k_0(N)\sim N/5$ such that two subsystems of $k$ particles each typically share entanglement if $k>k_0$, and typically do not share entanglement if $k<k_0$. Our methods work also for multipartite systems and for ``unbalanced'' systems such as $\C^{d_1} \otimes \C^{d_2} $, ${d_1} \neq {d_2} $. The arguments rely on random matrices, classical convexity, high-dimensional probability and geometry of Banach spaces; some of the auxiliary results may be of reference value.

The Orlicz–Legendre ellipsoids, which are in the framework of emerging dual Orlicz Brunn–Minkowski theory, are introduced for the first time. They are in some sense dual to the recently found Orlicz–John ellipsoids, and have largely generalized the classical Legendre ellipsoid of inertia. Several new affine isoperimetric inequalities are established. The connection between the characterization of Orlicz– Legendre ellipsoids and isotropy of measures is demonstrated.

Necessary and sufficient conditions are given for the existence of solutions to the discrete Lp Minkowski problem for the critical case where 0 < p < 1.