Fair Partitioning by Straight Lines

The author discusses stability results in reconstructing a plane convex body from the knowledge, with uniform error epsilon , of its projections along straight lines. The a priori assumptions about convexity gives an L2 stability of order epsilon , which is an improvement on well known stability results of order epsilon alpha , 0< alpha <or=1/2, for smooth plane domains.

We show that the maximum total perimeter of k plane convex bodies with disjoint interiors lying inside a given convex body C is equal to $\operatorname{per}\, (C)+2(k-1)\operatorname{diam}\, (C)$ , in the case when C is a square or an arbitrary triangle. A weaker bound is obtained for general plane convex bodies. As a consequence, we establish a bound on the perimeter of a polygon with at most k reflex angles lying inside a given plane convex body.

We show that every planar convex body is contained in a quadrangle whose area is less than ( 1 − 2.6 ⋅ 10 − 7 ) 2 times the area of the original convex body, improving the best known upper bound by W. Kuperberg.

Let C be a convex body in the Euclidean plane. By the relative distance of points p and q we mean the ratio of the Euclidean distance of p and q to the half of the Euclidean length of a longest chord of C parallel to pq. In this note we find the least upper bound of the minimum pairwise relative distance of six points in a plane convex body.

The relative distance of points a and b (or the relative length of the line-segment ab) in a convex body C is the ratio of the length of the line-segment ab to the half of the length of a longest chord of C parallel to ab. Langi conjectured that there exists no plane convex body whose boundary contains nine points at pairwise relative distances greater than \({4\sin\frac{\pi}{18} = 0.69459\ldots}\). In this paper we disprove this conjecture. Moreover, we find an infinite sequence of positive integers n for which there is a convex n-gon with no relatively long sides.

On fencing problems

Preface. Minimax theorems and their proofs S. Simons. A survey on minimax trees and associated algorithms C.G. Diderich, M. Gengler. An iterative method for the minimax problem L. Qi, W. Sun. A dual and interior point approach to solve convex min-max problems J.F. Sturm, S. Zhang. Determining the performance ratio of algorithm MULTIFIT for scheduling F. Cao. A study of on-line scheduling two-stage shops B. Chen, G.J. Woeginger. Maximin formulation of the apportionment of seats to parliament T. Helgason, et al. On shortest k-edge connected Steiner networks with rectilinear distance D.F. Hsu, et al. Mutually repellant sampling S.-H. Teng. Geometry and local optimality conditions for bilevel programs with quadratic strictly convex lower levels L.N. Vicente, P.H. Calamai. On the spherical one-center problem G. Xue, S. Sun. On min-max optimization of a collection of classical discrete optimization problems G. Xu, P. Kouvelis. Heilbronn problem for six points in a planar convex body A.W.M. Dress, et al. Heilbronn problem for seven points in a planar convex body L. Yang, Z. Zeng. On the complexity of min-max optimization problems and their approximation K.-I Ko, C.L. Lin. A competitive algorithm for the counterfeit coin problem X.-D. Hu, F.K. Hwang. A minimax alphabeta relaxation for global optimization J. Gu. Minimax problems in combinatorial optimization F. Cao, et al. Author index.

A (2-dimensional) double convex body 2K is a surface homeomorphic to the sphere consisting of two planar isometric compact convex bodies, K and K��, with boundaries glued in the obvious way. In this note we prove that, if K admits two perpendicular axes of symmetry and bdK satis es a certain curvature condition, then 2K admits an acute triangulation of size 72. In particular, each double ellipse admits such a triangulation.

On the relative distances of nine or ten points in the boundary of a plane convex body

On the relative distances of eleven points in the boundary of a plane convex body

In recent years P. C. Hammer's problem [8] of determining a convex body from its 'X-ray pictures' was investigated by Gardner and McMullen [4], Gardner [3], Falconer [2] and Volcic [15]. An earlier result is due to Giering [5]. An X-ray picture of a convex body in a direction may be identified with its Steiner symmetral in that direction. Some of these papers consider X-ray pictures taken from points not on the line at infinity, but here we are not concerned with that situation. Gardner and McMullen proved that there exist four directions such that the corresponding X-ray pictures distinguish between all convex bodies, and that no three directions can do this. Giering proved that, given a plane convex body K, there exist three directions depending on K, such that the corresponding X-ray pictures distinguish K from any other convex body. He has also shown that two directions are in general not enough. Convex bodies with the same X-ray pictures as a given one were called 'ghosts' in [14], in analogy with the ghost densities from computerized tomography [12]. It should be remembered that in the fundamental case of parallel rays from two orthogonal directions, besides a few triangular or quadrangular examples by Giering [6] and a rather obvious construction which basically interchanges two diagonally opposite, symmetrical pieces with two other diagonally opposite congruent pieces (diagonals of a rectangle), no deeper insight into the soul of a ghost of a convex body has been won. We are—as a consequence—far away from being able to characterize convex are not ghosts! (Note the equivalence between having and being a ghost!) Thus we In this situation, the question about the generic behaviour of convex bodies with regard to their ghosts appears interesting, but looks at a first glance, in view of the lack of knowledge described above, rather hopeless. However, in this paper we establish the validity of the (more comfortable?) assertion that most convex bodies are not ghosts! (Note the equivalence between having and being a ghost!) Thus we confirm a conjecture of the first author, motivated by the symmetries described above and also present in his examples from [14]. It is clear that the orthogonality of the two considered directions is unessential, because of the afrlne character of our problem. When we state it we do so just to fix the ideas. As a main open problem there remains the characterization of those convex bodies which are uniquely determined by two X-ray pictures. The analogous problem for measurable sets has been solved by Lorentz [11]. The space # of all convex curves in 1R, like the space & of all convex bodies in U, equipped with the Hausdorff distance S is a Baire space. 'Most ' means 'all, except those in a set of first category'. For a survey on properties of most convex bodies, see [16].

For a two-dimensional convex body, the Kovner-Besicovitch measure of symmetry is defined as the volume ratio of the largest centrally symmetric body contained inside the body to the original body. A classical result states that the Kovner-Besicovitch measure is at least $2/3$ for every convex body and equals $2/3$ for triangles. Lassak showed that an alternative measure of symmetry, i.e., symmetry about a line (axiality) has a value of at least $2/3$ for every convex body. However, the smallest known value of the axiality of a convex body is around $0.81584$, achieved by a convex quadrilateral. We show that every plane convex body has axiality at least $\frac{2}{41}(10 + 3 \sqrt{2}) \approx 0.69476$, thereby establishing a separation with the central symmetry measure. Moreover, we find a family of convex quadrilaterals with axiality approaching $\frac{1}{3}(\sqrt{2}+1) \approx 0.80474$. We also establish improved bounds for a ``folding" measure of axial symmetry for plane convex bodies. Finally, we establish improved bounds for a generalization of axiality to high-dimensional convex bodies.

The covariogram gK(x) of a convex body K ⊆ Ed is the function which associates to each x ∈ Ed the volume of the intersection of K with K + x, where Ed denotes the Euclidean d-dimensional space. Matheron (1986) asked whether gK determines K, up to translations and reflections in a point. Positive answers to Matheron's question have been obtained for large classes of planar convex bodies, while for d ≥ 3 there are both positive and negative results. One of the purposes of this paper is to sharpen some of the known results on Matheron's conjecture indicating how much of the covariogram information is needed to get the uniqueness of determination. We indicate some subsets of the support of the covariogram, with arbitrarily small Lebesgue measure, such that the covariogram, restricted to those subsets, identifies certain geometric properties of the body. These results are more precise in the planar case, but some of them, both positive and negative ones, are proved for bodies of any dimension. Moreover some results regard most convex bodies, in the Baire category sense. Another purpose is to extend the class of convex bodies for which Matheron's conjecture is confirmed by including all planar convex bodies possessing two nondegenerate boundary arcs being reflections of each other.

This paper studies a new Whitney type inequality on a compact domain $\Omega \subset {\mathbb R}^d$ that takes the form $$ \begin{align*} \inf_{Q\in \Pi_{r-1}^d(\mathcal{E})} \|f-Q\|_p \leq C(p,r,\Omega) \omega_{\mathcal{E}}^r(f,\mathrm{diam}(\Omega))_p,\ \ r\in {\mathbb N},\ \ 0&lt;p\leq \infty, \end{align*} $$ where $\omega _{\mathcal {E}}^r(f, t)_p$ denotes the rth order directional modulus of smoothness of $f\in L^p(\Omega )$ along a finite set of directions $\mathcal {E}\subset \mathbb {S}^{d-1}$ such that $\mathrm {span}(\mathcal {E})={\mathbb R}^d$ , $\Pi _{r-1}^d(\mathcal {E}):=\{g\in C(\Omega ):\ \omega ^r_{\mathcal {E}} (g, \mathrm {diam} (\Omega ))_p=0\}$ . We prove that there does not exist a universal finite set of directions $\mathcal {E}$ for which this inequality holds on every convex body $\Omega \subset {\mathbb R}^d$ , but for every connected $C^2$ -domain $\Omega \subset {\mathbb R}^d$ , one can choose $\mathcal {E}$ to be an arbitrary set of d independent directions. We also study the smallest number $\mathcal {N}_d(\Omega )\in {\mathbb N}$ for which there exists a set of $\mathcal {N}_d(\Omega )$ directions $\mathcal {E}$ such that $\mathrm {span}(\mathcal {E})={\mathbb R}^d$ and the directional Whitney inequality holds on $\Omega $ for all $r\in {\mathbb N}$ and $p&gt;0$ . It is proved that $\mathcal {N}_d(\Omega )=d$ for every connected $C^2$ -domain $\Omega \subset {\mathbb R}^d$ , for $d=2$ and every planar convex body $\Omega \subset {\mathbb R}^2$ , and for $d\ge 3$ and every almost smooth convex body $\Omega \subset {\mathbb R}^d$ . For $d\ge 3$ and a more general convex body $\Omega \subset {\mathbb R}^d$ , we connect $\mathcal {N}_d(\Omega )$ with a problem in convex geometry on the X-ray number of $\Omega $ , proving that if $\Omega $ is X-rayed by a finite set of directions $\mathcal {E}\subset \mathbb {S}^{d-1}$ , then $\mathcal {E}$ admits the directional Whitney inequality on $\Omega $ for all $r\in {\mathbb N}$ and $0&lt;p\leq \infty $ . Such a connection allows us to deduce certain quantitative estimate of $\mathcal {N}_d(\Omega )$ for $d\ge 3$ . A slight modification of the proof of the usual Whitney inequality in literature also yields a directional Whitney inequality on each convex body $\Omega \subset {\mathbb R}^d$ , but with the set $\mathcal {E}$ containing more than $(c d)^{d-1}$ directions. In this paper, we develop a new and simpler method to prove the directional Whitney inequality on more general, possibly nonconvex domains requiring significantly fewer directions in the directional moduli.

Let C m be a subset of a planar convex body C cut off by a straight line m, which remains in C after folding it along m. The maximum masf(C) of the ratio of the double area of C m to the area of C over all straight lines m is a measure of axial symmetry of C. We prove that \({{{\rm masf}}(P) > \frac{1}{2}}\) for every parallelogram P and that this inequality cannot be improved.