Embedding SnS^n into Rn+1R^{n+1} with given integral Gauss curvature and optimal mass transport on SnS^n

In his book on Convex Polyhedra (section 7.2), A.D. Aleksandrov raised a general question of finding variational statements and proofs of existence of polytopes with given geometric data. The first goal of this paper is to give a variational solution to the problem of existence and uniqueness of a closed convex hypersurface in Euclidean space with prescribed integral Gauss curvature. Our solution includes the case of a convex polytope. This problem was also first considered by Aleksandrov and below it is referred to as Aleksandrov's problem. The second goal of this paper is to show that in variational form the Aleksandrov problem is closely connected with the theory of optimal mass transport on a sphere with cost function and constraints arising naturally from geometric considerations

Optical design of two-reflector systems, the Monge-Kantorovich mass transfer problem and Fermat's principle

It is shown that the problem of designing a two-reflector system transforming a plane wave front with given intensity into an output plane front with prescribed output intensity can be formulated and solved as the Monge-Kantorovich mass transfer problem.Comment: 25 pages, 2 figure

Existence of optimal maps in the reflector-type problems

In this paper, we consider probability measures $\mu$ and $\nu$ on a $d$--dimensional sphere in \Rd, d \geq 1, and cost functions of the form c(\x,\y)=l(\frac{|\x-\y|^2}{2}) that generalize those arising in geometric optics where $l(t)=-\log t.$ We prove that if $\mu$ and $\nu$ vanish on $(d-1)$--rectifiable sets, if $|l'(t)|>0,$ $\lim_{t\to 0^+}l(t)=+\infty,$ and $g(t):=t(2-t)(l'(t))^2$ is monotone then there exists a unique optimal map $T_o$ that transports $\mu$ onto $\nu,$ where optimality is measured against $c.$ Furthermore, \inf_{\x}|T_o\x-\x|>0. Our approach is based on direct variational arguments. In the special case when $l(t)=-\log t,$ existence of optimal maps on the sphere was obtained earlier by Glimm-Oliker and independently by X.-J. Wang under more restrictive assumptions. In these studies, it was assumed that either $\mu$ and $\nu$ are absolutely continuous with respect to the $d$--dimensional Haussdorff measure, or they have disjoint supports. Another aspect of interest in this work is that it is in contrast with a result by Gangbo-McCann who proved that when $l(t)=t$ then existence of an optimal map fails when $\mu$ and $\nu$ are supported by Jordan surfaces

• SummaryDiffraction and Geometry Statement of the problem

A principal problem in design of antenna systems mounted on platforms such aircraft, satellites, and ships is to determine the optimal location of antennas on the platform so that the desired radiation coverage is achieved. Similarly, the problem of optimal location of antennas arises when multiple antennas are mounted on the platform and and mutual interference must be minimized. The documentation supplied by antenna manufacturers contains a description of antenna pattern when the antenna is operating in space or on a ground plane. Diffraction and Geometry Statement of the problem 3 In fact, when antenna is mounted on a platform, such as a modern aircraft, the net antenna pattern is impacted dramatically by the complex geometry of the platform. Consequently, to ensure proper functioning of an antenna system mounted on a platform it is critically important to determine the effects due to the platform. Diffraction and Geometry Possible Approaches 4 To determine, at least approximately, the actual antenna performance the following approaches have been used: Diffraction and Geometr