A variational formula for the Lutwak affine surface areas j of convex bodies in Rn is established when 1 ≤ j ≤ n − 1. By using introduced new ellipsoids associated with projection functions of convex bodies, we prove a sharp isoperimetric inequality for j , which opens up a new passage to attack the longstanding Lutwak conjecture in convex geometry.

In this paper we investigate the one-dimensional hyperbolic mean curvature flow for closed plane curves. More precisely, we consider a family of closed curves S 1[0, T) 2 which satisfies the following evolution equation 2 F t 2 (u, t)= k (u, t) N(u, t)- (u, t),(u, t) S 1[0, T) with the initial data F (u, 0)= F 0 (u) a n d F t (u, 0)= f (u) N 0, where k is the mean curvature and is the unit inner normal vector of the plane curve F (u, t), f (u) and are the initial velocity and the unit inner normal vector of the initial convex closed curve Fo, respectively, and is given by ( 2 F s t, F t) T, in which stands for the unit tangent vector. The above problem is an initial value problem for a system of partial differential equations for F, it can be completely reduced to an initial value problem for a single partial differential equation for its support function. The latter equation is a hyperbolic Monge-Ampere

Sufficient conditions are given for the existence of solutions to the discrete LpMinkowski problem for p-capacity when 0 <p <1and 1 <p <2.

In this paper we consider proper holomorphic embeddings of finitely connected planar domains into ℂ^{$n$}that approximate given proper embeddings on smooth curves. As a side result we obtain a tool for approximating a $\mathcal{C}^{\infty}$ diffeomorphism on a polynomially convex set in ℂ^{$n$}by an automorphism of ℂ^{$n$}that satisfies some additional properties along a real embedded curve.

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