Equilateral triangles on a Jordan curve and a generalization of a theorem of Dold

Let be a Jordan curve in the plane. It is a simple topological riddle to determine if there is an equilateral triangle with vertices on γ. By reformulating this question in the paradigm of configuration spaces and test maps, we can solve this riddle using a Borsuk-Ulam type theorem obtained using equivariant methods.     

Every Jordan curve inscribes uncountably many rhombi

Geometriae Dedicata - We prove that every Jordan curve in $$mathbb {R}^2$$ inscribes uncountably many rhombi. No regularity condition is assumed on the Jordan curve.     

A remark on translates of a box spline

We prove an express theorem on the stability of integer translates of a box spline and clear proposition 4. 10. 2 in [6] for box spline case. Also, we provide a way to construct a linear projection from L_p to the space of integer translates of a box spline. Consequently, the equivalence among the stability, global linear independence and local linear independence of translates of a box spile follows. Partially supported by ARO contract No. DAAL 4-87-0025     

Riemann boundary-value problem on an open rectifiable Jordan curve. II

The Riemann boundary-value problem is solved for the classes of open rectifiable Jordan curves extended as compared with previous results and functions defined on these curves.     

An elementary geometric nonstandard proof of the Jordan curve theorem

We give an elementary proof, using nonstandard analysis, of the Jordan curve theorem. We also give a nonstandard generalization of the theorem. The proof is purely geometrical in character, without any use of topological concepts and is based on a discrete finite form of the Jordan theorem, whose proof is purely combinatorial.Some familiarity with nonstandard analysis is assumed. The rest of the paper is self-contained except for the proof a discrete standard form of the Jordan theorem. The proof is based on hyperfinite approximations to regions on the plane.Research of the first author partially supported by FONDECYT Grant # 91-1208 and of the second author, by FONDECYT Grant # 90-0647.     

Analyticity on families of circles

It is known that iff is a continuous function on the complex plane which extends holomorphically from each circle surrounding the origin, thef is not necessarily holomorphic. In the paper we prove that if, in addition,f extends holomorphically from each circle belonging to an open family of circles which do not surround the origin, thef is holomorphic.     

Point transversals to translates of a trapezoid

Anm-transversal to a family of convex sets in the plane is anm-point set which intersects every members of the family. One of Grübaum’s conjectures says that a planar family of translates of a convex compact set has a 3-transversal provided that any two of its members intersect. Recently the conjecture has been proved affirmatively (see [4]). In the present paper we provide a different and straightforward proof for the conjecture for the family of translates of a closed trapezoid in the plane and give several concrete 3-transversals.     

The fundamentality of translates of a continuous function on spheres

In this paper we discuss the fundamentality of translates of a continuous function on the unit spheres of Euclidean spaces. Our result partially answers a question of Cheney and Xu [1].     

Analyticity of functions analytic on circles

Let Δ be the open unit disc in, let pbΔ, and let f be a continuous function on which extends holomorphically from each circle in centered at the origin and from each circle in which passes through p. Then f is holomorphic on Δ.     

Transversal numbers of translates of a convex body

Let F be a family of translates of a fixed convex set M in Rⁿ. Let τ(F) and ν(F) denote the transversal number and the independence number of F, respectively. We show that ν(F)?τ(F)?8ν(F)-5 for n=2 and τ(F)?2^n-1nⁿν(F) for n?3. Furthermore, if M is centrally symmetric convex body in the plane, then ν(F)?τ(F)?6ν(F)-3.     

Covering a Polish group by translates of a nowhere dense set

We show that, for every nonlocally compact Polish group G with a left-invariant complete metric ρ, we have covG=cov(M). Here, covG is the minimal number of translates of a fixed closed nowhere dense subset of G, which is needed to cover G, and cov(M) is the minimal cardinality of a cover of the real line R by meagre sets.     

Analyticity of nonlinear semigroups

The Cauchy problemdu/dt =Au(t),u(0) =u ₀∈D(A) has analytic solutions whenA has first and second Gateaux derivatives along the solution curve in a certain weak sense. HereA is a maximal monotone operator in a complex Hilbert space.     

Lattice translates of a polytope and the Frobenius problem

This paper considers the Frobenius problem: Givenn natural numbersa ₁,a ₂,...a _n such that their greatest common divisor is 1, find the largest natural number that is not expressible as a nonnegative integer combination of them. This problem can be seen to be NP-hard. For the casesn=2,3 polynomial time algorithms, are known to solve it. Here a polynomial time algorithm is given for every fixedn. This is done by first proving an exact relation between the Frobenius problem and a geometric concept called the covering radius. Then a polynomial time algorithm is developed for finding the covering radius of any polytope in a fixed number of dimensions. The last algorithm relies on a structural theorem proved here that describes for any polytopeK, the setK+ ^h ={xx ⁿ ;x=y+z;yK;z ⁿ } which is the portion of space covered by all lattice translates ofK. The proof of the structural theorem relies on some recent developments in the Geometry of Numbers. In particular, it uses a theorem of Kannan and Lovász [11], bounding the width of lattice-point-free convex bodies and the techniques of Kannan, Lovász and Scarf [12] to study the shapes of a polyhedron obtained by translating each facet parallel, to itself. The concepts involved are defined from first principles. In a companion paper [10], I extend the structural result and use that to solve a general problem of which the Frobenius problem is a special case.Supported by NSF-Grant CCR 8805199