Exceptional ergodic directions in Eaton lenses

We construct examples of ergodic vortical flows in periodic configurations of Eaton lenses of fixed radius. We achieve this by studying a family of infinite translation surfaces that are ?²-covers of slit tori. We show that the Hausdorff dimension of lattices for which the vertical flow is ergodic is bigger than 3/2. Moreover, the lattices are explicitly constructed.     

Sets convex in a cone of directions

We consider sets which are convex in directions from some cone K. We generalize some well-known properties of ordinary convex sets for the case of K-convex sets and give some applications in optimization theory.     

Isoperimetric quotients for a decomposed convex body

In the Euclidean plane, decompose a convex body T into ngeq 2 convex bodies T ₁ ,ldots ,T _n with areas also denoted by T ₁ ,ldots ,T _n , and with perimeters L ₁ ,ldots ,L _n . For T a polygon with at most six sides, G. Fejes Tóth and also L. Fejes Tóth showed that the isoperimetric quotient (L ₁ + ⋅s + L _n )/(sqrt T ₁ + ⋅s + sqrt T _n ) is greater than the corresponding isoperimetric quotient of a regular hexagon if T _i /T _j for any i, j is bounded from below by some appropriate constant. We generalize this result to any convex body T , and we show the analogous result for the isoperimetric quotient (L ² ₁ + ⋅s + L ² _n )/(T ₁ + ⋅s + T _n ) . Received April 21, 1999, and in revised form June 21, 2000. Online publication January 17, 2001.     

On a conjugate directions method for solving strictly convex QP problem

Problem of solving the strictly convex, quadratic programming problem is studied. The idea of conjugate directions is used. First we assume that we know the set of directions conjugate with respect to the hessian of the goal function. We apply n simultaneous directional minimizations along these conjugate directions starting from the same point followed by the addition of the directional corrections. Theorem justifying that the algorithm finds the global minimum of the quadratic goal function is proved. The way of effective construction of the required set of conjugate directions is presented. We start with a vector with zero value entries except the first one. At each step new vector conjugate to the previously generated is constructed whose number of nonzero entries is larger by one than in its predecessor. Conjugate directions obtained by means of the above construction procedure with appropriately selected parameters form an upper triangular matrix which in exact computations is the Cholesky factor of the inverse of the hessian matrix. Computational cost of calculating the inverse factorization is comparable with the cost of the Cholesky factorization of the original second derivative matrix. Calculation of those vectors involves exclusively matrix/vector multiplication and finding an inverse of a diagonal matrix. Some preliminary computational results on some test problems are reported. In the test problems all symmetric, positive definite matrices with dimensions from \(14\times 14\) to \(2000\times 2000\) from the repository of the Florida University were used as the hessians.     

A variable-penalty alternating directions method for convex optimization

We study a generalized version of the method of alternating directions as applied to the minimization of the sum of two convex functions subject to linear constraints. The method consists of solving consecutively in each iteration two optimization problems which contain in the objective function both Lagrangian and proximal terms. The minimizers determine the new proximal terms and a simple update of the Lagrangian terms follows. We prove a convergence theorem which extends existing results by relaxing the assumption of uniqueness of minimizers. Another novelty is that we allow penalty matrices, and these may vary per iteration. This can be beneficial in applications, since it allows additional tuning of the method to the problem and can lead to faster convergence relative to fixed penalties. As an application, we derive a decomposition scheme for block angular optimization and present computational results on a class of dual block angular problems. This material is based on research supported by the Air Force Office of Scientific Research Grant AFOSR-89-0410 and by NSF Grants CCR-8907671, CDA-9024618 and CCR-9306807.     

On the minimal convex annulus of a planar convex body

In this paper we introduce the notion of a minimal convex annulusK (C) of a convex bodyC, generalizing the concept of a minimal circular annulus. Then we prove the existence — as for the minimal circular annulus — of a Radon partition of the set of contact points of the boundaries ofK (C) andC. Subsequently, the uniqueness ofK (C) is shown. Finally, it is proven that, for typicalC, the boundary ofC has precisely two points in common with each component of the boundary ofK (C).     

Minimal annulus of a convex body

It is my pleasant duty to thank Professor P. Gruber for his suggestions.     

Inscribed simplexes of a convex body

This paper advances the theorem that for any smooth point M of the boundary of a convex body K Rⁿ there exists a nondegenerate simplex inscribed in K with a vertex at M that is similar to a given n-dimensional simplex. Similar problems are considered and unanswered questions are posed.Translated from Ukrainskii Geometricheskii Sbornik, No. 35, pp. 47–49, 1992.     

On the inner parallel body of a convex body

It is shown that the set of boundary points of a convex body at which there are no interior tangent balls of positive radius has zero surface area.     

How to hold a convex body?

Convex bodies are often used for mathematical tests. They occasionally try to escape. Can the testing mathematician hold them still by using a circle? Rarely not.