On the horseshoe conjecture for maximal distance minimizers

We estimate the maximal number of integral points which can be on a convex arc in $${\mathbb {R}}^2$$ with given length, minimal radius of curvature and initial slope.

The aim of this note is to prove that any compact metric space can be made connected at a minimal cost, where the cost is taken to be the one-dimensional Hausdorff measure.

We study a shape optimization problem for the first eigenvalue of an elliptic operator in divergence form, with nonconstant coefficients, over a fixed domain $\Omega$. Dirichlet conditions are imposed along $\partial \Omega$ and, additionally, along a set $\Sigma$ of prescribed length (one-dimensional Hausdorff measure). We look for the best shape and position for the supplementary Dirichlet region $\Sigma$ in order to maximize the first eigenvalue. The limit distribution of the optimal sets, as their prescribed length tends to infinity, is characterized via $\Gamma$-convergence of suitable functionals defined over varifolds; the use of varifolds, as opposed to probability measures, allows one to keep track of the local orientation of the optimal sets (which comply with the anisotropy of the problem), and not just of their limit distribution.

Second-order energies on thin structures: variational theory and non-local effects

On partial regularity for weak solutions to the Navier–Stokes equations

k-Dimensional intersections of convex sets and convex kernels

On the equivalence of extrapolation and Richardson’s iteration and its applications

Article Free Access Share on Voronoi diagrams and containment of families of convex sets on the plane Authors: M. Abellanas Universidad Politécnica de Madrid, Spain Universidad Politécnica de Madrid, SpainView Profile , G. Hernandez Universidad Politécnica de Madrid, Spain Universidad Politécnica de Madrid, SpainView Profile , R. Klein Fern Universität Hagen, Germany Fern Universität Hagen, GermanyView Profile , V. Neumann-Lara Universidad Nacional Autonoma de México, Mexico Universidad Nacional Autonoma de México, MexicoView Profile , J. Urrutia University of Ottawa, Canada University of Ottawa, CanadaView Profile Authors Info & Claims SCG '95: Proceedings of the eleventh annual symposium on Computational geometrySeptember 1995 Pages 71–78https://doi.org/10.1145/220279.220287Published:01 September 1995Publication History 3citation259DownloadsMetricsTotal Citations3Total Downloads259Last 12 Months12Last 6 weeks4 Get Citation AlertsNew Citation Alert added!This alert has been successfully added and will be sent to:You will be notified whenever a record that you have chosen has been cited.To manage your alert preferences, click on the button below.Manage my AlertsNew Citation Alert!Please log in to your account Save to BinderSave to BinderCreate a New BinderNameCancelCreateExport CitationPublisher SiteeReaderPDF

Let X X be a separable Banach space. We give an almost characterization of the existence of a Finite Dimensional Decomposition (FDD for short) for X X in terms of Lipschitz retractions onto generating compact subsets K K of X X . In one direction, if X X admits an FDD then we construct a Lipschitz retraction onto a small generating convex and compact set K K . On the other hand, we prove that if X X admits a “small” generating compact Lipschitz retract then X X has the π \pi -property. It is still unknown if the π \pi -property is isomorphically equivalent to the existence of an FDD. For dual Banach spaces this is true, so our results give a characterization of the FDD property for dual Banach spaces X X . We give an example of a small generating convex compact set which is not a Lipschitz retract of C [ 0 , 1 ] C[0,1] , although it is contained in a small convex Lipschitz retract and contains another one. We characterize isomorphically Hilbertian spaces as those Banach spaces X X for which every convex and compact subset is a Lipschitz retract of X X . Finally, we prove that a convex and compact set K K in any Banach space with a Uniformly Rotund in Every Direction norm is a uniform retract, of every bounded set containing it, via the nearest point map.

Hyperspaces of convex bodies of constant width

Article A Helly-type theorem for hyperplane transversals to well-separated convex sets Share on Authors: Boris Aronov Polytechnic University, Brooklyn, NY Polytechnic University, Brooklyn, NYView Profile , Jacob E. Goodman City College, City University of New York, New York, NY City College, City University of New York, New York, NYView Profile , Richard Pollack Courant Institute of Mathematical Sciences, New York University, New York, NY Courant Institute of Mathematical Sciences, New York University, New York, NYView Profile , Rephael Wenger The Ohio State University, Columbus, OH The Ohio State University, Columbus, OHView Profile Authors Info & Claims SCG '00: Proceedings of the sixteenth annual symposium on Computational geometryMay 2000 Pages 57–63https://doi.org/10.1145/336154.336178Online:01 May 2000Publication History 0citation233DownloadsMetricsTotal Citations0Total Downloads233Last 12 Months4Last 6 weeks1 Get Citation AlertsNew Citation Alert added!This alert has been successfully added and will be sent to:You will be notified whenever a record that you have chosen has been cited.To manage your alert preferences, click on the button below.Manage my AlertsNew Citation Alert!Please log in to your account Save to BinderSave to BinderCreate a New BinderNameCancelCreateExport CitationPublisher SiteGet Access

In this paper the problem of finding the error of an approximate solution for ill-posed problems on some compact sets is discussed. Sets of bounded monotonic or convex functions and sets of functions with a given Lipschitz constant are considered. These functions are given on a segment. Two approaches are used to construct sets of approximate solutions. In the first approach we assume that a function for the right-hand side of an operator equation is given on the whole segment. In the second approach we use only grid values of this function. Applying these approaches, we obtain quadratic or linear programming problems and find functions bounding above and below these approximate solutions. In the paper we use two algorithms to find errors that determine the sets of approximate solutions for the cases when operators of the problems are positive or integral. The inverse problem for the heat conduction equation is considered as a model example.

The following problem is discussed: Find a (constrained or unconstrained) minimizer of the function $$ F(x) = \\max _{y\\in G_1}\\min _{z\\in G_2} \\varphi (x,y,z),$$ where } ( x,y,z ) is a function defined and continuous on $ R^n\ imes R^p\ imes R^q, G_1\\subset R^p $ and $ G_2\\subset R^q $ are compact sets in the respective spaces. The case where the function } is continuously differentiable was studied earlier. It is well known that the problem of minimizing the function F is nonconvex and multiextremal. It is shown in the article that the problem is reduced to solving a family of minimax problems. The discrete case (where G 1 and G 2 contain a finite number of points) is discussed in more detail. In such a case the problem is reduced to solving a finite number of minimax problems. A necessary condition for a point to be a global minimizer and a sufficient condition for a point to be a local one are proved. Numerical methods for finding stationary points (i.e. points satisfying the necessary condition) are proposed. These methods (as well as the necessary condition) are of nonlocal nature thus in some cases allowing one to escape from a local minimizer. Each stationary point is a local minimizer while the converse is not true.

It is proved that if the nonempty intersection of bounded closed convex sets A \ B is contained in (A u F) ( (B u F) and one of the following holds true: (i) the space X is less-than-three dimensional, (ii) A ( B is convex, (iii) F is a one-point set, then A \ B A u F or A \ B B u F (Theorems 2 and 3). Moreover, under some hypotheses the characterization of A and B such that A \ B is a summand of A ( B is given (Theorem 3). In this paper we consider the inclusion A\B (AuF)((B uF) which plays a central role in proving criterion of convex minimality of pairs of bounded closed convex sets (12), (13). Minimal pairs of compact convex sets were investigated in a series of papers (2), (4), (5), (10), (12), and others. Pairs of compact convex sets naturally arise in quasidierential

Newton method for solving a class of smooth convex programming