On the ground state eigenfunction of a convex domain in Euclidean space

In 1913, E. Helly[i] has established: When C 1 , ..., C m are convex domains in an n-dimensional Euclidean space and every set of n+I of these domains have a common point, then there exists a point which is common to all the domains. Recently[6] this theorem has obtained an enhanced importance because of the numerous applications which it affords.

In this paper we consider Monge–Ampère equations on compact Hessian manifolds, or equivalently Monge–Ampère equations on certain unbounded convex domains in Euclidean space, with a periodicity constraint given by the action of an affine group. In the case where the affine group action is volume preserving, i.e., when the manifold is special, the solvability of the corresponding Monge–Ampère equation was first established by Cheng and Yau using the continuity method. In the general case we set up a variational framework involving certain dual manifolds and a generalization of the classical Legendre transform. We give existence and uniqueness results and elaborate on connections to optimal transport and quasi-periodic tilings of convex domains.

We consider a queue where the server is the Euclidean space, and the customers are random closed sets (RACSs) of the Euclidean space. These RACSs arrive according to a Poisson rain and each of them has a random service time (in the case of hail falling on the Euclidean plane, this is the height of the hailstone, whereas the RACS is its footprint). The Euclidean space serves customers at speed 1. The service discipline is a hard exclusion rule: no two intersecting RACSs can be served simultaneously and service is in the first-in–first-out order, i.e. only the hailstones in contact with the ground melt at speed 1, whereas the others are queued. A tagged RACS waits until all RACSs that arrived before it and intersecting it have fully melted before starting its own melting. We give the evolution equations for this queue. We prove that it is stable for a sufficiently small arrival intensity, provided that the typical diameter of the RACS and the typical service time have finite exponential moments. We also discuss the percolation properties of the stationary regime of the RACS in the queue.

Here we analyze three dimensional analogues of the classic Crofton formula for planar compact convex sets. In this formula a fundamental role is played by the visual angle of the convex set from an exterior point. A generalization of the visual angle to convex sets in the Euclidean space is the visual solid angle. This solid angle, being an spherically convex set in the unit sphere, has perimeter, area and other geometric quantities to be considered. The main goal of this note is to express invariant quantities of the original convex set depending on volume, surface area and mean curvature integral by means of integrals of functions related to the solid angle.

Introduction. This paper gives counterexamples for two unrelated conjectures which pertain to convexity theory. Supposef is any function defined on a closed convex set C in finitedimensional Euclidean space. Its envelope function env f is defined as the pointwise supremum of all linear functions which f dominates everywhere. It is elementary and well known that env f is convex, and hence continuous in the relative interior of C. It is similarly known that env f is lower-semicontinuous on the boundary of C. Witsenhausen [14] uses and studies envelope functions. He suggested the following orally: CONJECTURE 1. Iff is continuous everywhere on C, then envf is also continuous everywhere on C. I provide a very simple counterexample in 3-space which shows that this is not true. Incidentally, though the proof is not on paper, I feel fairly certain that Witsenhausen's conjecture is correct in 2-space. If C is a closed convex set in finite-dimensional Euclidean space, then for any point x in the space there is a unique point P(x) in C which is nearest to x. The mapping P is called the projection onto C. (Many interesting results about this mapping are given in Phelps [9], [10].) It is easy to prove and well known that P is a contraction mapping, hence continuous. It is also easy to see that P need not have much in the way of differentiability properties, for if C is a polyhedron in 2-space, then it is easy to find points and directions at which P does not even have a two-sided directional derivative. The existence in general of a one-sided directional derivative, however, is not so easy to decide. This question does not seem to be raised in such natural places as Eggleston [4] and Valentine [11]. CONJECTURE 2. For all points and directions, P has a one-sided directional derivative. My mildly complicated counterexample is a set C (in 3-space), which is the convex hull of a countable infinity of points. I suspect that the conjecture is correct in 2-space. In n-space it would be interesting to know whether the set of points and directions at which the conjecture holds is dense in EnXSn-. If not, the convex set would have to be extremely jagged!

The method of projections onto convex sets to find a point in the intersection of a finite number of closed convex sets in a Euclidean space, may lead to slow convergence of the constructed sequence when that sequence enters some narrow “corridor” between two or more convex sets. A way to leave such corridor consists in taking a big step at different moments during the iteration, because in that way the monotoneous behaviour that is responsible for the slow convergence may be interrupted. In this paper we present a technique that may introduce interruption of the monotony for a sequential algorithm, but that at the same time guarantees convergence of the constructed sequence to a point in the intersection of the sets. We compare experimentally the behaviour concerning the speed of convergence of the new algorithm with that of an existing monotoneous algorithm.

Moments of the measure of the intersection of a given set with the union of random sets may be called moments of coverage. These moments specified for the case of independent random sets in the Euclidean space, were treated by several authors. Kolmogorov (1933) and Robbins (1944), (1945), (1947) derived transformation formulas for the moments of coverage under some specified conditions. Robbins' formula was applied by Robbins (1945), Santaló (1947) and Garwood (1947) to computations of expectations and variances of coverage in the cases of spheres and intervals in 2-dimensional and in n-dimensional Euclidean space. The computations were carried out under the assumption of a simple distribution function for the centers of the covering figures (a homogeneous distribution in most of the cases and a normal one in some of them). Neyman and Bronowski (1945) computed by a different method the variance of coverage for some specific cases in the Euclidean plane.

A bosonic Laplacian is a conformally invariant second‐order differential operator acting on smooth functions defined on domains in Euclidean space and taking values in higher‐order irreducible representations of the special orthogonal group. In this paper, we firstly introduce the motivation for study of the generalized Maxwell operators and bosonic Laplacians (also known as the higher spin Laplace operators). Then, with the help of connections between Rarita–Schwinger type operators and bosonic Laplacians, we solve Poisson's equation for bosonic Laplacians. A representation formula for bounded solutions to Poisson's equation in Euclidean space is also provided. In the end, we provide Green's formulas for bosonic Laplacians in scalar‐valued and Clifford‐valued cases, respectively. These formulas reveal that bosonic Laplacians are self‐adjoint with respect to a given L2 inner product on certain compact supported function spaces.

Using the first exit time for Brownian motion from a smoothly bounded domain in Euclidean space, we define two natural functionals on the space of embedded, compact, oriented, unparametrized hypersurfaces in Euclidean space. We develop explicit formulas for the first variation of each of the functionals and characterize the critical points.

The purpose of this paper is to investigate the topological and analytical restrictions on a domain D in euclidean n-space R n on which an infinite discrete quasiconformal group can act. We will see that the restrictions are indeed severe, unlike the case of a discrete group of topological or differentiable homeomorphisms.KeywordsUnit BallLimit PointUniversal CoverTangent PlaneQuasiconformal MappingThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

The Borsuk number of a set S of diameter d >0 in Euclidean n-space is the smallest value of m such that S can be partitioned into m sets of diameters less than d. Our aim is to generalize this notion in the following way: The k-fold Borsuk number of such a set S is the smallest value of m such that there is a k-fold cover of S with m sets of diameters less than d. In this paper we characterize the k-fold Borsuk numbers of sets in the Euclidean plane, give bounds for those of centrally symmetric sets, smooth bodies and convex bodies of constant width, and examine them for finite point sets in the Euclidean 3-space.

Separating collections of points in Euclidean spaces

This is an overview of some of the similarities and differences between structures such as frameworks and cabled tensegrities in the hyperbolic plane and hyperbolic space on the one hand and the Euclidean plane, the sphere and Euclidean space on the other hand.KeywordsHyperbolic planeconfigurationsframeworktensegrity

We are studying a characteristic property of the disc in the Euclidean plane ℝ^2 with the help of an intersection property of the semicircles with end-points in the boundary of the disc and a characteristic property of the ball in the Euclidean space ℝ^3.

Isoperimetric inequalities for quermassintegrals