Optimizing Area and Perimeter of Convex Sets for Fixed Circumradius and Inradius

 In this paper two problems posed by Santaló are solved: we determine the planar convex sets which have maximum and minimum area or perimeter when the circumradius and the inradius are given, obtaining complete systems of inequalities for the cases (A, R, r) and (p, R, r). This work is supported in part by Dirección General de Investigación (MCYT) BFM2001-2871, and by OTKA grants No 31984 and 30012 Received October 15, 2001; revised January 29, 2002     

Inradius and Circumradius of Various Convex Cones Arising in Applications

This note addresses the issue of computing the inradius and the circumradius of a convex cone in a Euclidean space. It deals also with the related problem of finding the incenter and the circumcenter of the cone. We work out various examples of convex cones arising in applications.     

Perimeter, Diameter and Area of Convex Sets in the Hyperbolic Plane

The paper studies the relation between the asymptotic valuesof the ratios area/length (F/L) and diameter/length (D/L) ofa sequence of convex sets expanding over the whole hyperbolicplane. It is known that F/L goes to a value between 0 and 1depending on the shape of the contour. In the paper, it is firstof all seen that D/L has limit value between 0 and 1/2 in strongcontrast with the euclidean situation in which the lower boundis 1/ (D/L = 1/ if and only if the convex set has constant width).Moreover, it is shown that, as the limit of D/L approaches 1/2,the possible limit values of F/L reduce. Examples of all possiblelimits F/L and D/L are given.     

平面凸域周长公式的初等证明

文给出了平面凸域周长公式的初等证明     

The Mean Perimeter of Some Random Plane Convex Sets Generated by a Brownian Motion

If C ₁ is the convex hull of the curve of a standard Brownian motion in the complex plane watched from 0 to 1, we consider the convex hulls of C ₁ and several rotations of it and compute the mean of the length of their perimeter by elementary calculations. This can be seen geometrically as a study of the exit time by a Brownian motion from certain polytopes having the unit circle as an inscribed one.     

A Continuation Method for Solving Fixed Point Problems in Unbounded Convex Sets

In this paper, an unbounded condition is presented, under which we are able to utilize the interior point homotopy method to solve the Brouwer fixed point problem on unbounded sets. Two numerical examples in R3 are presented to illustrate the results in this paper.     

The Smallest Convex Cover for Triangles of Perimeter Two

We find the unique smallest convex region in the plane that contains a congruent copy of every triangle of perimeter two. It is the triangle ABC with AB=2/3, B=60°, and BC1.00285.     

Separable Determination of the Fixed Point Property of Convex Sets in Banach Spaces

In this paper, we first show that for every mapping $f$ from a metric space $Ω$ to itself which is continuous off a countable subset of $Ω,$ there exists a nonempty closed separable subspace $S ⊂ Ω$ so that $f|_S$ is again a self mapping on $S.$ Therefore, both the fixed point property and the weak fixed point property of a nonempty closed convex set in a Banach space are separably determined. We then prove that every separable subspace of $c_0(\Gamma)$ (for any set $\Gamma$) is again lying in $c_0.$ Making use of these results, we finally presents a simple proof of the famous result: Every non-expansive self-mapping defined on a nonempty weakly compact convex set of $c_0(\Gamma)$ has a fixed point.     

Canonical Theorems for Convex Sets

Let F be a family of pairwise disjoint compact convex sets in the plane such that none of them is contained in the convex hull of two others, and let r be a positive integer. We show that F has r disjoint ⌊ c _r n⌋ -membered subfamilies F _i (1 ≤ i ≤ r) such that no matter how we pick one element F _i from each F _i , they are in convex position, i.e., every F _i appears on the boundary of the convex hull of ⋃ _i=1 ^r F _i . (Here c _r is a positive constant depending only on r .) This generalizes and sharpens some results of Erdős and Szekeres, Bisztriczky and Fejes Tóth, Bárány and Valtr, and others. <lsiheader> <onlinepub>26 June, 1998 <editor>Editors-in-Chief: &lsilt;a href=../edboard.html#chiefs&lsigt;Jacob E. Goodman, Richard Pollack&lsilt;/a&lsigt; <pdfname>19n3p427.pdf <pdfexist>yes <htmlexist>no <htmlfexist>no <texexist>yes <sectionname> </lsiheader> Received April 30, 1997, and in revised form August 5, 1997.     

Self-Concordant Barriers for Convex Approximations of Structured Convex Sets

We show how to approximate the feasible region of structured convex optimization problems by a family of convex sets with explicitly given and efficient (if the accuracy of the approximation is moderate) self-concordant barriers. This approach extends the reach of the modern theory of interior-point methods, and lays the foundation for new ways to treat structured convex optimization problems with a very large number of constraints. Moreover, our approach provides a strong connection from the theory of self-concordant barriers to the combinatorial optimization literature on solving packing and covering problems.     

Klee Theorem for Linearly Convex Sets

We prove a complex analog of the classical Klee theorem for strongly linearly convex closed sets.     

A Ramsey-Type Result for Convex Sets

Given a family of n convex compact sets in the plane, one canalways choose n of them which are either pairwise disjoint orpairwise intersecting. On the other hand, there exists a familyof n segments in the plane such that the maximum size of a subfamilywith pairwise disjoint or pairwise intersecting elements inn^log2/log5 n^0·431.     

Well-positioned Closed Convex Sets and Well-positioned Closed Convex Functions

We characterize the class of those closed convex sets which have a barrier cone with a nonempty interior. As a consequence, we describe the set of those proper extended-real-valued functionals for which the domain of their Fenchel conjugate has a nonempty interior. As an application, we study the stability of the solution set of a semi-coercive variational inequality.     

Contraction and Expansion of Convex Sets

Let S{\mathcal{S}} be a set system of convex sets in ℝ ^d . Helly’s theorem states that if all sets in S{\mathcal{S}} have empty intersection, then there is a subset S￠ ì S{\mathcal{S}}'\subset{\mathcal{S}} of size d+1 which also has empty intersection. The conclusion fails, of course, if the sets in S{\mathcal{S}} are not convex or if S{\mathcal{S}} does not have empty intersection. Nevertheless, in this work we present Helly-type theorems relevant to these cases with the aid of a new pair of operations, affine-invariant contraction, and expansion of convex sets. These operations generalize the simple scaling of centrally symmetric sets. The operations are continuous, i.e., for small ε>0, the contraction C ^−ε and the expansion C ^ε are close (in the Hausdorff distance) to C. We obtain two results. The first extends Helly’s theorem to the case of set systems with nonempty intersection:     

Triangles of Minimal Perimeter Circumscribed about a Convex n-Gon

An algorithm for finding a circumscribed triangle of the minimal perimeter is suggested. Properties of such a triangle are described. Bibliography: 1 title.     

Geometric Condition Measures and Smoothness Condition Measures for Closed Convex Sets and Linear Regularity of Infinitely Many Closed Convex Sets

In this paper, we study geometric condition measures and smoothness condition measures of closed convex sets, bounded linear regularity, and linear regularity. We show that, under certain conditions, the constant for the linear regularity of infinitely many closed convex sets can be characterized by the geometric condition measure of the intersection or by the smoothness condition measure of the intersection. We study also the bounded linear regularity and present some interesting properties of the general linear regularity problem.The author is grateful to the referees for valuable and constructive suggestions. In particular, she thanks a referee for drawing her attention to Corollary 5.14 of Ref. 3, which inspired her to derive Theorem 4.2 and Corollary 4.2 in the revision of this paper.     

New Combinations of Convex Sets

The family of convex sets in a (finite dimensional) real vector space admits several unary and binary operations – dilatation, intersection, convex hull, vector sum – which preserve convexity. These generalize to convex functions, where there are in fact further operations of this kind. Some of the latter may be regarded as combinations of two such operations, acting on complementary subspaces. In this paper, a general theory of such mixed operations is introduced, and some of its consequences developed.     

Weakly Convex Sets and Their Properties

In this paper, the notion of a weakly convex set is introduced. Sharp estimates for the weak convexity constants of the sum and difference of such sets are given. It is proved that, in Hilbert space, the smoothness of a set is equivalent to the weak convexity of the set and its complement. Here, by definition, the smoothness of a set means that the field of unit outward normal vectors is defined on the boundary of the set; this vector field satisfies the Lipschitz condition. We obtain the minimax theorem for a class of problems with smooth Lebesgue sets of the goal function and strongly convex constraints. As an application of the results obtained, we prove the alternative theorem for program strategies in a linear differential quality game.     

A Helly-Type Theorem for Unions of Convex Sets

We prove that for any d, k ≥ 1 there are numbers q = q(d,k) and h = h(d,k) such that the following holds: Let be a family of subsets of the d-dimensional Euclidean space, such that the intersection of any subfamily of consisting of at most q sets can be expressed as a union of at most k convex sets. Then the Helly number of is at most h. We also obtain topological generalizations of some cases of this result. The main result was independently obtained by Alon and Kalai, by a different method. Received April 14, 1995, and in revised form August 1, 1995.     

Sets of Finite Perimeter and the Gauss-Green Theorem with Singularities

We establish the n-dimensional divergence theorem in a formreaching the limits of generality with respect to the geometricaland the analytical assumptions.