A Small Cover for Convex Unit Arcs

An arc in the plane is convex if it is simple (i.e., one–one except that its endpoints may coincide) and lies on the boundary of its convex hull. We describe a compact convex plane set of area about 0.2466 that contains a congruent copy of each convex plane arc of unit length, a reduction of about 1.1% from the smallest such set previously known.     

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.     

Covers for closed curves of length two

The least area α ₂ of a convex set in the plane large enough to contain a congruent copy of every closed curve of length two lies between 0.385 and 0.491, as has been known for more than 38 years. We improve these bounds by showing that 0.386 < α ₂ < 0.449.     

The smallest triangular cover for triangles of diameter one

A convex region covers a family of curves if it contains a congruent copy of each curve in the family, and a “worm problem” for that family is to find the convex region of smallest area. In this paper, we find the smallest triangular cover of any prescribed shape for the familyS of all triangles of diameter 1.     

Covering with Fat Convex Discs

According to a theorem of L. Fejes Tóth [4], if non-crossing congruent copies of a convex disc K cover a convex hexagon H, then the density of the discs relative to H is at least area K/f_K(6) where f_K(6) denotes the maximum area of a hexagon contained in K. We say that a convex disc is r-fat if it is contained in a unit circle C and contains a concentric circle c of radius r. Recently, Heppes [7] showed that the above inequality holds without the non-crossing assumption if K is a 0.8561-fat ellipse. We show that the non-crossing assumption can be omitted if K is an r₀-fat convex disc with r₀ = 0.933 or an r₁-fat ellipse with r₁ = 0.741.     

On the existence ofp-units and minkowski units in totally real cyclic fields

LetK be a totally real cyclic number field of degree n > 1. A unit inK is called an m-unit, if the index of the group generated by its conjugations in the group U*_K of all units modulo ±1 is coprime tom. It is proved thatK contains an m-unit for every m coprime to n. The mutual relationship between the existence of m-units and the existence of a Minkowski unit is investigated for those n for which the class number hФ(ζ_n) of the n-th cyclotomic field is equal to 1. For n which is a product of two distinct primes p and q, we derive a sufficient condition for the existence of a Minkowski unit in the case when the field K contains a p-unit for every prime p, namely that every ideal contained in a finite list (see Lemma 11) is principal. This reduces the question of whether the existence of a p-unit and a q-unit implies the existence of a Minkowski unit to a verification of whether the above ideals are principal. As a corollary of this, we establish that every totally real cyclic field K of degree n = 2q, where q = 2, 3 or 5, contains a Minkowski unit if and only if it contains a 2-unit and a q-unit.     

The Empty Hexagon Theorem

Let P be a finite set of points in general position in the plane. Let C(P) be the convex hull of P and let Cⁱ_P be the ith convex layer of P. A minimal convex set S of P is a convex subset of P such that every convex set of P ∩ C(S) different from S has cardinality strictly less than |S|. Our main theorem states that P contains an empty convex hexagon if C¹_P is minimal and C⁴_P is not empty. Combined with the Erdos-Szekeres theorem, this result implies that every set P with sufficiently many points contains an empty convex hexagon, giving an affirmative answer to a question posed by Erdos in 1977.     

Convex Bodies Instead of Needless in Buffon's Experiment

In this paper we consider the two events that a random congruent copy of a convex body meets each one of two given families of equidistant lines in the plane. The probabilities are easily calculated. Then it is discovered that there always exists a value for the angle between the nonparallel lines, such that the two events be independent. For convex bodies of constant width, and only for them, the two events are independent for any .     

Some unavoidable subdigraphs of tournaments

Let H(n, i) be a simple (n ? 1)-path v₁ → v₂ → …? → v_n with an additional arc v₁v_i (3 ? i ? n). We prove that for each n and i (3 ? i ? n), with few exceptions, every n-tournament T_n contains a copy of H(n, i).     

Extreme differences between weakly open subsets and convex combinations of slices in Banach spaces

We show that every Banach space containing isomorphic copies of _c0$c_0$ can be equivalently renormed so that every nonempty relatively weakly open subset of its unit ball has diameter 2 and, however, its unit ball still contains convex combinations of slices with diameter arbitrarily small, which improves in an optimal way the known results about the size of this kind of subsets in Banach spaces.     

Ball-Polyhedra

We study two notions. One is that of spindle convexity. A set of circumradius not greater than one is spindle convex if, for any pair of its points, it contains every short circular arc of radius at least one, connecting them. The other objects of study are bodies obtained as intersections of finitely many balls of the same radius, called ball-polyhedra. We find analogues of several results on convex polyhedral sets for ball-polyhedra.     

Stability Constants in Linear Spaces

We investigate the stability constants of convex sets in linear spaces. We prove that the stability constants of affinity and of the Jensen equation are of the same order of magnitude for every convex set in arbitrary linear spaces, even for functions mapping into an arbitrary Banach space. We also show that the second Whitney constant corresponding to the bounded functions equals half of the stability constant of the Jensen equation whenever the latter is finite. We show that if a convex set contains arbitrarily long segments in every direction, then its Jensen and Whitney constants are uniformly bounded. We prove a result that reduces the investigation of the stability constants to the case when the underlying set is the unit ball of a Banach space. As an application we prove that if D is convex and every δ-Jensen function on D differs from a Jensen function by a bounded function, then the stability constants of D are finite.     

多部竞赛图中包含某条弧的圈

多部竞赛图或n部竞赛图是指一个完全n部无向图的定向图.2007年Volkmann证明了每个强连通的n部竞赛图(n≥3)至少存在一条弧它包含在从3到n的每个长度的圈中.在此基础上给出了强连通n部竞赛图中存在一条弧它包含在从3到n+1的每个长度的圈中的一个充分条件,并举例说明该条件在某种意义上的最佳可能性.     

Empty Convex Hexagons in Planar Point Sets

Erdős asked whether every sufficiently large set of points in general position in the plane contains six points that form a convex hexagon without any points from the set in its interior. Such a configuration is called an empty convex hexagon. In this paper, we answer the question in the affirmative. We show that every set that contains the vertex set of a convex 9-gon also contains an empty convex hexagon.     

On convex bodies containing a given number of lattice points

Given a convex body K in Euclidean space, a necessary and sufficient condition is established in order that for each n there exists a homothetic copy of K containing exactly n lattice points. Similar theorems are proved for congruent copies of K and for discrete sets other than lattices.     

Weakly compact operators onH ∞

We prove that a weakly compact operator fromH or any of its even duals into an arbitrary Banach space is uniformly convexifying. By using this, we establish three dicothomies: (1) every operator defined onH or any of its even duals either fixes a copy ofl or factors through a Banach space having the Banach-Saks property; (2) every quotient ofH or any of its even duals either contains a copy ofl or is super-reflexive; (3) every subspace ofL ¹/H ₀ ¹ or any of its even duals either contains a complemented copy ofl ¹ or is super-reflexive.     

Induced trees in graphs of large chromatic number

Gyárfás and Sumner independently conjectured that for every tree T and integer k there is an integer f(k, T) such that every graph G with χ(G) > f(k, t) contains either K_k or an induced copy of T. We prove a ‘topological’ version of the conjecture: for every tree T and integer k there is g(k,T) such that every graph G with χ(G) > g(k,t) contains either K_k or an induced copy of a subdivision of T. © 1997 John Wiley & Sons, Inc.     

Isomorphic copies of ℓ1 for m‐homogeneous non‐analytic Bohnenblust–Hille polynomials

We employ a classical result by Toeplitz (1913) and the seminal work by Bohnenblust and Hille on Dirichlet series (1931) to show that the set of continuous m‐homogeneous non‐analytic polynomials on c ₀ contains an isomorphic copy of ℓ₁. Moreover, we can have this copy of ℓ₁ in such a way that every non‐zero element of it fails to be analytic at precisely the same point.     

The structure of 4‐strong tournaments containing exactly three out‐arc pancyclic vertices

Yao et al. (Discrete Appl Math 99 (2000), 245–249) proved that every strong tournament contains a vertex u such that every out‐arc of u is pancyclic and conjectured that every k‐strong tournament contains k such vertices. At present, it is known that this conjecture is true for k = 1, 2, 3 and not true for k?4. In this article, we obtain a sufficient and necessary condition for a 4‐strong tournament to contain exactly three out‐arc pancyclic vertices, which shows that a 4‐strong tournament contains at least four out‐arc pancyclic vertices except for a given class of tournaments. Furthermore, our proof yields a polynomial algorithm to decide if a 4‐strong tournament has exactly three out‐arc pancyclic vertices.     

关于格点覆盖的一个注记

对于一个给定椭球,本文给出了它的任一全等椭球都包含一个整点的一个充分必要条件.