Approximation in the mean by convex functions

We give elementary proofs for the existence and uniqueness of the best L₁-approximation to a continuous function from the class of convex functions on a closed interval, and describe thebest approximation in terms of certain piecewise linear functions.     

Approximation of convex set-valued functions

Approximation of set-valued functions is introduced and discussed under a convexity assumption. In particular, a theorem on positive linear operators is given.     

Calculus of convex polyhedra and polyhedral convex functions by utilizing a multiple objective linear programming solver

ABSTRACT

The article deals with operations defined on convex polyhedra or polyhedral convex functions. Given two convex polyhedra, operations like Minkowski sum, intersection and closed convex hull of the union are considered. Basic operations for one convex polyhedron are, for example, the polar, the conical hull and the image under affine transformation. The concept of a P-representation of a convex polyhedron is introduced. It is shown that many polyhedral calculus operations can be expressed explicitly in terms of P-representations. We point out that all the relevant computational effort for polyhedral calculus consists in computing projections of convex polyhedra. In order to compute projections we use a recent result saying that multiple objective linear programming (MOLP) is equivalent to the polyhedral projection problem. Based on the MOLP solver bensolve a polyhedral calculus toolbox for Matlab and GNU Octave is developed. Some numerical experiments are discussed.     

Approximation of convex bodies from chord functions

In this paper we describe a method for computing derivatives of the polar representations of the boundary of a convex body, from generalized chord functions at two points.     

Approximation of functions regular in convex polygons by exponential polynomials

Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 40, No. 4, pp. 446–452, July–August, 1988.     

Refold rigidity of convex polyhedra

We show that every convex polyhedron may be unfolded to one planar piece, and then refolded to a different convex polyhedron. If the unfolding is restricted to cut only edges of the polyhedron, we identify several polyhedra that are “edge-refold rigid” in the sense that each of their unfoldings may only fold back to the original. For example, each of the 43,380 edge unfoldings of a dodecahedron may only fold back to the dodecahedron, and we establish that 11 of the 13 Archimedean solids are also edge-refold rigid. We begin the exploration of which classes of polyhedra are and are not edge-refold rigid, demonstrating infinite rigid classes through perturbations, and identifying one infinite nonrigid class: tetrahedra.     

On weakly convex star-shaped polyhedra

Weakly convex polyhedra which are star-shaped with respect to one of their vertices are infinitesimally rigid. This is a partial answer to the question as to whether every decomposable weakly convex polyhedron is infinitesimally rigid. The proof is based on a recent result of Izmestiev on the geometry of convex caps.     

A conjecture on convex polyhedra

We select a class of pyramids of a particular shape and propose a conjecture that precisely these pyramids are of greatest surface area among the closed convex polyhedra having evenly many vertices and the unit geodesic diameter. We describe the geometry of these pyramids. The confirmation of our conjecture will solve the “doubly covered disk” problem of Alexandrov. Through a connection with Reuleaux polygons we prove that on the plane the convex n-gon of unit diameter, for odd n, has greatest area when it is regular, whereas this is not so for even n.     

Closed covers of compact convex polyhedra

Further generalizations of the Knaster-Kuratowski-Mazurkiewicz type theorems are presented. Also given is a new proof of David Gale's theorem on a family ofn closed covers of a simplex.The authors would like to thank Richard P. McLean for his valuable suggestions. A part of this paper was completed during Tatsuro Ichiishi's visit to Instytut Podstaw Informatyki, Polskiej Akademii Nauk (Polish Academy of Sciences), Warsaw, Poland, in early September 1990. Financial supports from the Polish Academy of Sciences and from the Ohio State University are gratefully acknowledged.     

Approximation of convex bodies by rectangles

For every plane convex body there is a pair of inscribed and circumscribed homothetic rectangles. The positive ratio of homothety is not greater than 2.Research supported in part by Komitet Badan Naukowych (Committee of Scientific Research), grant number 2 2005 92 03.     

Approximation of convex bodies by simplices

This paper deals with the following kind of approximation of a convex bodyQ in Euclidean space E ⁿ by simplices: which is the smallest positive numberh _S(Q) such thatS ₁ Q S ₂ for a simplexS ₁ and its homothetic copyS ₂ of ratioh _S(Q). It is shown that ifS ₀ is a simplex of maximal volume contained inQ, then a homothetic copy ofS ₀ of ratio 13/3 containsQ.     

Approximation of convex discs by polygons

We consider the class of all convex discs with areas and perimeters bounded by given constants. Which disc of this class has the least possible area deviation from ak-gon? This and related questions are the subject of the present paper.     

Approximation of convex bodies by polytopes

LetC be a convex body ofE ^d and consider the symmetric difference metric. The distance ofC to its best approximating polytope having at mostn vertices is 0 (1/n ^2/(d?1)) asn→∞. It is shown that this estimate cannot be improved for anyC of differentiability class two. These results complement analogous theorems for the Hausdorff metric. It is also shown that for both metrics the approximation properties of «most» convex bodies are rather irregular and that ford=2 «most» convex bodies have unique best approximating polygons with respect to both metrics.     

Approximation of convex sets by polytopes

The survey contains results related to different aspects of polyhedral approximation of convex bodies and some adjacent problems. __________ Translated from Sovremennaya Matematika. Fundamental’nye Napravleniya (Contemporary Mathematics. Fundamental Directions), Vol. 22, Geometry, 2007.     

On the enumeration of convex polyhedra

We state and justify an asymptotic formula for the number of combinatorially distinct convex polyhedra with 2n faces, all triangular.