scholarly journals On Maximal $S$-Free Sets and the Helly Number for the Family of $S$-Convex Sets

2013 ◽  
Vol 27 (3) ◽  
pp. 1610-1624 ◽  
Author(s):  
Gennadiy Averkov
Keyword(s):  
Convex Sets ◽  
Helly Number ◽  
The Family ◽  
Free 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.

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 A helly number for unions of two boxes inR2

1985 ◽  
Vol 8 (2) ◽  
pp. 267-273 ◽  
Author(s):  
Marilyn Breen
Keyword(s):  
Convex Sets ◽  
Helly Number ◽  
Boundary Points ◽  
Polygonal Region

LetSbe a polygonal region in the plane with edges parallel to the coordinate axes. If every5or fewer boundary points ofScan be partitioned into setsAandBso thatconv A⋃ conv B⫅S, thenSis a union of two convex sets, each a rectangle. The number5is best possible.Without suitable hypothesis on edges ofS, the theorem fails. Moreover, an example reveals that there is no finite Helly number which characterizes arbitrary unions of two convex sets, even for polygonal regions in the plane.

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.

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

Maximal $S$-Free Convex Sets and the Helly Number

2016 ◽  
Vol 30 (4) ◽  
pp. 2206-2216 ◽  
Author(s):  
Michele Conforti ◽  
Marco Di Summa
Keyword(s):  
Convex Sets ◽  
Helly Number

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.

On the complexity of the family of convex sets in ℝ d

2016 ◽  
Vol 99 (3-4) ◽  
pp. 534-544
Author(s):  
V. V. Pernay
Keyword(s):  
Convex Sets ◽  
The Family

scholarly journals Semicontinuous functions and convex sets in C(K) spaces

2007 ◽  
Vol 82 (1) ◽  
pp. 111-121 ◽  
Author(s):  
J. P. Moreno
Keyword(s):  
Banach Space ◽  
Compact Space ◽  
Convex Sets ◽  
Minkowski Addition ◽  
Stability Properties ◽  
Compact Spaces ◽  
The Family ◽  
The Stability ◽  
Open Question ◽  
Semicontinuous Functions

AbstractThe stability properties of the family ℳ of all intersections of closed balls are investigated in spaces C(K), where K is an arbitrary Hausdorff compact space. We prove that ℳ is stable under Minkowski addition if and only if K is extremally disconnected. In contrast to this, we show that ℳ is always ball stable in these spaces. Finally, we present a Banach space (indeed a subspace of C[0, 1]) which fails to be ball stable, answering an open question. Our results rest on the study of semicontinuous functions in Hausdorff compact spaces.

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