Minimum recession-compatible subsets of closed convex sets

A subset B of a closed convex set A is recession-compatible with respect to A if A can be expressed as the Minkowski sum of B and the recession cone of A. We show that if A contains no line, then there exists a recession-compatible subset of A that is minimal with respect to set inclusion. The proof only uses basic facts of convex analysis and does not depend on Zorn’s Lemma. An application of this result to the error bound theory in optimization is presented.     

The perimeter of rounded convex planar sets

A convex set is inscribed into a rectangle with sides a and 1/a so that the convex set has points on all four sides of the rectangle. By “rounding” we mean the composition of two orthogonal linear transformations parallel to the edges of the rectangle, which makes a unit square out of the rectangle. The transformation is also applied to the convex set, which now has the same area, and is inscribed into a square. One would expect this transformation to decrease the perimeter of the convex set as well. Interestingly, this is not always the case. For each a we determine the largest and smallest possible increase of the perimeter.      

A better decomposition theorem for simply connected m-convex sets

A set S in R^d is said to be m-convex, m ? 2, if and only if for every m points in S, at least one of the line segments determined by these points lies in S. For S a closed m-convex set in R², various decomposition theorems have been obtained to express S as a finite union of convex sets. However, the previous bounds may be lowered further, and we have the following result:In case S is simply connected, then S is a union of σ(m) or fewer convex sets, where $\text{σ(m) = [(m ?}\text{N}\text{)(m ?}\text{3}\text{2}\text{) +}\text{3}\text{2}\text{]}$.Moreover, this result induces an improved decomposition in the general case as well.     

Some remarks on convex surfaces in simply connected homogeneous three manifolds

Here we extend Hadamards’ Theorem to other homogeneous three manifolds, i.e., we prove that a compact orientable immersed surface Σ in general position whose principal curvatures κ _i , i = 1, 2, satisfy κ _i ≥ τ, is an embedded sphere.     

Disjoint empty convex pentagons in planar point sets

In this paper we obtain the first non-trivial lower bound on the number of disjoint empty convex pentagons in planar points sets. We show that the number of disjoint empty convex pentagons in any set of n points in the plane, no three on a line, is at least $\left\lfloor {\tfrac{{5n}} {{47}}} \right\rfloor $ . This bound can be further improved to $\tfrac{{3n - 1}} {{28}} $ for infinitely many n.     

On convex embeddings of planar 3‐connected graphs

A well‐known Tutte's theorem claims that every 3‐connected planar graph has a convex embedding into the plane. Tutte's arguments also show that, moreover, for every nonseparating cycle C of a 3‐connected graph G, there exists a convex embedding of G such that C is a boundary of the outer face in this embedding. We give a simple proof of this last result. Our proof is based on the fact that a 3‐connected graph admits an ear assembly having some special properties with respect to the nonseparating cycles of the graph. This fact may be interesting and useful in itself. © 2000 John Wiley & Sons, Inc. J. Graph Theory 33: 120–124, 2000     

Non-uniqueness of the reconstruction for connected and simply connected sets in the plane by their fixed finite projections

We discuss a problem to reconstruct the measurable sets in the plane from their fixed finite projections. In the main theorem, we construct an example of connected and simply connected polygons which are not uniquely reconstructed by their fixed finite projections. We also make a comparison between our main theorem and the known results on this problem.     

Tilting simply connected algebras

Let A be a representation-finite simply connected algebra, and T_Abe an arbitrary tilting module (in the sense of Happei and Ringel). We show that B = End T_A satisfies the separation condition of Bautista, Larrion and Salmeron. This implies that B is also simply connected.     

Critical simply connected algebras

It is well-known that simply connected algebras are uniquely determined by a graded tree.Reversely,each graded tree gives rise to a not necessarily representation-finite algebra. We call an algebra critical provided it is not representation-finite, but any proper convex full subalgebra is.All critical algebras arising from graded trees are classified.     

Estimates of covering numbers of convex sets with slowly decaying orthogonal subsets

Covering numbers of precompact symmetric convex subsets of Hilbert spaces are investigated. Lower bounds are derived for sets containing orthogonal subsets with norms of their elements converging to zero sufficiently slowly. When these sets are convex hulls of sets with power-type covering numbers, the bounds are tight. The arguments exploit properties of generalized Hadamard matrices. The results are illustrated by examples from machine learning, neurocomputing, and nonlinear approximation.     

Distance sets that are a shift of the integers and Fourier basis for planar convex sets

The aim of this paper is to prove that if a planar set A has a difference set Δ(A) satisfying Δ(A) ? ?⁺ + s for suitable s then A has at most 3 elements. This result is motivated by the conjecture that the disk has no more than 3 orthogonal exponentials. Further, we prove that if A is a set of exponentials mutually orthogonal with respect to any symmetric convex set K in the plane with a smooth boundary and everywhere non-vanishing curvature, then #(A ∩ [?q, q]²) ≦ C(K) _q where C(K) is a constant depending only on K. This extends and clarifies in the plane the result of Iosevich and Rudnev. As a corollary, we obtain the result from [8] and [9] that if K is a centrally symmetric convex body with a smooth boundary and non-vanishing curvature, then L ²(K) does not possess an orthogonal basis of exponentials.     

On semilocally simply connected spaces

The purpose of this paper is: (i) to construct a space which is semilocally simply connected in the sense of Spanier even though its Spanier group is non-trivial; (ii) to propose a modification of the notion of a Spanier group so that via the modified Spanier group semilocal simple connectivity can be characterized; and (iii) to point out that with just a slightly modified definition of semilocal simple connectivity which is sometimes also used in literature, the classical Spanier group gives the correct characterization within the general class of path-connected topological spaces.While the condition “semilocally simply connected” plays a crucial role in classical covering theory, in generalized covering theory one needs to consider the condition “homotopically Hausdorff” instead. The paper also discusses which implications hold between all of the abovementioned conditions and, via the modified Spanier groups, it also unveils the weakest so far known algebraic characterization for the existence of generalized covering spaces as introduced by Fischer and Zastrow. For most of the implications, the paper also proves the non-reversibility by providing the corresponding examples. Some of them rely on spaces that are newly constructed in this paper.     

h-Cobordisms between simply connected 4-manifolds

h-cobordisms between simply connected 4-manifolds are studied. It is shown that most inertial h-cobordisms have a handle decomposition with one 2-handle and one 3-handle, and h-cobordisms between nondiffeomorphic manifolds have handle decompositions with the minimal number of handles consistent with a diffeomorphism between the stabilized ends. Also the number of distinct h-cobordisms between two fixed manifolds is described in terms of isomorphisms of their quadratic forms. These results are applied to Dolgachev surfaces and the Kummer surface using recent work of Donaldson, Friedman and Morgan, and Matumoto.     

Derived tubular strongly simply connected algebras

Let be a finite dimensional algebra over an algebraically closed field . Assume for a connected quiver and an admissible ideal of . We study algebras which are derived equivalent to tubular algebras. If is strongly simply connected and has more than six vertices, then is derived tubular if and only if (i) the homological quadratic form is a non-negative of corank two and (ii) no vector of is orthogonal (with respect tho the homological bilinear form) to the radical of .