The Demyanov metric and some other metrics in the family of convex sets

2012 ◽  
Vol 10 (6) ◽  
Author(s):  
Tadeusz Rzeżuchowski
Keyword(s):  
Convex Sets ◽  
Hausdorff Metric ◽  
Facial Structure ◽  
The Family

AbstractWe describe some known metrics in the family of convex sets which are stronger than the Hausdorff metric and propose a new one. These stronger metrics preserve in some sense the facial structure of convex sets under small changes of sets.

scholarly journals Further steps on the reconstruction of convex polyominoes from orthogonal projections

2021 ◽  
Author(s):  
Paolo Dulio ◽  
Andrea Frosini ◽  
Simone Rinaldi ◽  
Lama Tarsissi ◽  
Laurent Vuillon
Keyword(s):  
Discrete Geometry ◽  
Convex Subset ◽  
Convex Sets ◽  
A Priori ◽  
Orthogonal Projections ◽  
Reconstruction Process ◽  
Combinatorial Properties ◽  
The Family ◽  
Discrete Counterpart ◽  
Interior Part

AbstractA remarkable family of discrete sets which has recently attracted the attention of the discrete geometry community is the family of convex polyominoes, that are the discrete counterpart of Euclidean convex sets, and combine the constraints of convexity and connectedness. In this paper we study the problem of their reconstruction from orthogonal projections, relying on the approach defined by Barcucci et al. (Theor Comput Sci 155(2):321–347, 1996). In particular, during the reconstruction process it may be necessary to expand a convex subset of the interior part of the polyomino, say the polyomino kernel, by adding points at specific positions of its contour, without losing its convexity. To reach this goal we consider convexity in terms of certain combinatorial properties of the boundary word encoding the polyomino. So, we first show some conditions that allow us to extend the kernel maintaining the convexity. Then, we provide examples where the addition of one or two points causes a loss of convexity, which can be restored by adding other points, whose number and positions cannot be determined a priori.

Čebyšev Sets in Hyperspaces over Rn

2009 ◽  
Vol 61 (2) ◽  
pp. 299-314 ◽  
Author(s):  
Robert J. MacG. Dawson and ◽  
Maria Moszyńska
Keyword(s):  
Metric Space ◽  
Convex Sets ◽  
Compact Convex ◽  
Hausdorff Metric ◽  
Linear Structure ◽  
Convex Compact ◽  
Nearest Neighbour ◽  
Compact Sets ◽  
Strictly Convex ◽  
Convex Compact Sets

Abstract. A set in a metric space is called a Čebyšev set if it has a unique “nearest neighbour” to each point of the space. In this paper we generalize this notion, defining a set to be Čebyšev relative to another set if every point in the second set has a unique “nearest neighbour” in the first. We are interested in Čebyšev sets in some hyperspaces over Rn, endowed with the Hausdorff metric, mainly the hyperspaces of compact sets, compact convex sets, and strictly convex compact sets. We present some new classes of Čebyšev and relatively Čebyšev sets in various hyperspaces. In particular, we show that certain nested families of sets are Čebyšev. As these families are characterized purely in terms of containment,without reference to the semi-linear structure of the underlyingmetric space, their properties differ markedly from those of known Čebyšev sets.

EQUI-ATTRACTION AND THE CONTINUOUS DEPENDENCE OF PULLBACK ATTRACTORS ON PARAMETERS

2004 ◽  
Vol 04 (03) ◽  
pp. 373-384 ◽  
Author(s):  
DESHENG LI ◽  
P. E. KLOEDEN
Keyword(s):  
Metric Space ◽  
Continuous Dependence ◽  
Hausdorff Metric ◽  
Regularity Conditions ◽  
Pullback Attractors ◽  
Compact Metric Space ◽  
Nonautonomous Dynamical Systems ◽  
The Family ◽  
Nonempty Compact ◽  
Autonomous Dynamical System

The equi-attraction properties of uniform pullback attractors [Formula: see text] of nonautonomous dynamical systems (θ,ϕλ) with a parameter λ∈Λ, where Λ is a compact metric space, are investigated; here θ is an autonomous dynamical system on a compact metric space P which drives the cocycle ϕλon a complete metric state space X. In particular, under appropriate regularity conditions, it is shown that the equi-attraction of the family [Formula: see text] uniformly in p∈P is equivalent to the continuity of the setvalued mappings [Formula: see text] in λ with respect to the Hausdorff metric on the nonempty compact subsets of X.

scholarly journals Continued fractions built from convex sets and convex functions

2015 ◽  
Vol 17 (05) ◽  
pp. 1550003 ◽  
Author(s):  
Ilya Molchanov
Keyword(s):  
Convex Functions ◽  
Continued Fractions ◽  
Convex Sets ◽  
Sufficient Conditions ◽  
Ordered Semigroup ◽  
Minkowski Addition ◽  
Partially Ordered ◽  
The Family ◽  
Sufficient Conditions For Convergence

In a partially ordered semigroup with the duality (or polarity) transform, it is possible to define a generalization of continued fractions. General sufficient conditions for convergence of continued fractions are provided. Two particular applications concern the cases of convex sets with the Minkowski addition and the polarity transform and the family of non-negative convex functions with the Legendre–Fenchel and Artstein-Avidan–Milman transforms.

scholarly journals On Mosco convergence of convex sets

1988 ◽  
Vol 38 (2) ◽  
pp. 239-253 ◽  
Author(s):  
Gerald Beer
Keyword(s):  
Convex Sets ◽  
Hausdorff Metric ◽  
Mosco Convergence ◽  
Natural Topology ◽  
Weakly Compact ◽  
Reflexive Space ◽  
Convergence Of Sequences ◽  
Metric Topology ◽  
Hausdorff Metric Topology

We present a natural topology compatible with the Mosco convergence of sequences of closed convex sets in a reflexive space, and characterise the topology in terms of the continuity of the distance between convex sets and fixed weakly compact ones. When the space is separable, the topology is Polish. As an application, we show that in this context, most closed convex sets are almost Chebyshev, a result that fails for the stronger Hausdorff metric topology.

Order cancellation law in the family of bounded convex sets

2019 ◽  
Vol 77 (2) ◽  
pp. 289-300
Author(s):  
J. Grzybowski ◽  
R. Urbański
Keyword(s):  
Convex Sets ◽  
The Family ◽  
Bounded Convex

scholarly journals A Purely Combinatorial Proof of the Hadwiger Debrunner $(p,q)$ Conjecture

10.37236/1316 ◽  
1996 ◽  
Vol 4 (2) ◽  
Author(s):  
N. Alon ◽  
D. J. Kleitman
Keyword(s):  
Convex Sets ◽  
Compact Convex ◽  
Nonempty Intersection ◽  
Combinatorial Proof ◽  
The Family

A family of sets has the $(p,q)$ property if among any $p$ members of the family some $q$ have a nonempty intersection. The authors have proved that for every $p \geq q \geq d+1$ there is a $c=c(p,q,d) < \infty$ such that for every family ${\cal F}$ of compact, convex sets in $R^d$ which has the $(p,q)$ property there is a set of at most $c$ points in $R^d$ that intersects each member of ${\cal F}$, thus settling an old problem of Hadwiger and Debrunner. Here we present a purely combinatorial proof of this result.

scholarly journals Diagonal Minkowski classes, zonoid equivalence, and stable laws

2020 ◽  
pp. 1950091
Author(s):  
Ilya Molchanov ◽  
Felix Nagel
Keyword(s):  
Integral Transform ◽  
Hausdorff Metric ◽  
Convex Bodies ◽  
Stable Laws ◽  
Symmetric Convex Body ◽  
Symmetric Convex ◽  
Support Functions ◽  
Cosine Transform ◽  
The Family ◽  
Minkowski Sums

We consider the family of convex bodies obtained from an origin symmetric convex body [Formula: see text] by multiplication with diagonal matrices, by forming Minkowski sums of the transformed sets, and by taking limits in the Hausdorff metric. Support functions of these convex bodies arise by an integral transform of measures on the family of diagonal matrices, equivalently, on Euclidean space, which we call [Formula: see text]-transform. In the special case, if [Formula: see text] is a segment not lying on any coordinate hyperplane, one obtains the family of zonoids and the cosine transform. In this case two facts are known: the vector space generated by support functions of zonoids is dense in the family of support functions of origin symmetric convex bodies; and the cosine transform is injective. We show that these two properties are equivalent for general [Formula: see text]. For [Formula: see text] being a generalized zonoid, we determine conditions that ensure the injectivity of the [Formula: see text]-transform. Relations to mixed volumes and to a geometric description of one-sided stable laws are discussed. The later probabilistic application motivates our study of a family of convex bodies obtained as limits of sums of diagonally scaled [Formula: see text]-balls.

A Class of Quasi-Nonexpansive Multi-Valued Maps

1975 ◽  
Vol 18 (5) ◽  
pp. 709-714 ◽  
Author(s):  
Chyi Shiau ◽  
Kok-Keong Tan ◽  
Chi Song Wong
Keyword(s):  
Metric Space ◽  
Hausdorff Metric ◽  
The Family

Let (X, d) be a (nonempty) metric space. bc(X) will denote the family of all nonempty bounded closed subsets of X endowed with the Hausdorff metric D induced by d [2, pp. 205]. Let/be a map of X into bc(X). f is nonexpansive at a point x in X if for all y in X.

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