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 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 A variational approach to the alternating projections method

2021 ◽  
Author(s):  
Carlo Alberto De Bernardi ◽  
Enrico Miglierina
Keyword(s):  
Convex Sets ◽  
Nonempty Intersection ◽  
Feasibility Problem ◽  
Alternating Projections ◽  
Limit Sets ◽  
Von Neumann ◽  
Convex Feasibility ◽  
Starting Point ◽  
Method Of Alternating Projections ◽  
Alternating Projections Method

AbstractThe 2-sets convex feasibility problem aims at finding a point in the nonempty intersection of two closed convex sets A and B in a Hilbert space H. The method of alternating projections is the simplest iterative procedure for finding a solution and it goes back to von Neumann. In the present paper, we study some stability properties for this method in the following sense: we consider two sequences of closed convex sets $$\{A_n\}$$ { A n } and $$\{B_n\}$$ { B n } , each of them converging, with respect to the Attouch-Wets variational convergence, respectively, to A and B. Given a starting point $$a_0$$ a 0 , we consider the sequences of points obtained by projecting on the “perturbed” sets, i.e., the sequences $$\{a_n\}$$ { a n } and $$\{b_n\}$$ { b n } given by $$b_n=P_{B_n}(a_{n-1})$$ b n = P B n ( a n - 1 ) and $$a_n=P_{A_n}(b_n)$$ a n = P A n ( b n ) . Under appropriate geometrical and topological assumptions on the intersection of the limit sets, we ensure that the sequences $$\{a_n\}$$ { a n } and $$\{b_n\}$$ { b n } converge in norm to a point in the intersection of A and B. In particular, we consider both when the intersection $$A\cap B$$ A ∩ B reduces to a singleton and when the interior of $$A \cap B$$ A ∩ B is nonempty. Finally we consider the case in which the limit sets A and B are subspaces.

scholarly journals An elementary renewal theorem for random compact convex sets

1995 ◽  
Vol 27 (4) ◽  
pp. 931-942 ◽  
Author(s):  
Ilya S. Molchanov ◽  
Edward Omey ◽  
Eugene Kozarovitzky
Keyword(s):  
Support Function ◽  
Convex Sets ◽  
Compact Convex ◽  
Renewal Function ◽  
Renewal Theorem ◽  
Closed Sets ◽  
Random Closed Sets ◽  
Minkowski Sums ◽  
Image Position ◽  
Unit Vectors

A set-valued analog of the elementary renewal theorem for Minkowski sums of random closed sets is considered. The corresponding renewal function is defined as where are Minkowski (element-wise) sums of i.i.d. random compact convex sets. In this paper we determine the limit of H(tK)/t as t tends to infinity. For K containing the origin as an interior point, where hK(u) is the support function of K and is the set of all unit vectors u with EhA(u) > 0. Other set-valued generalizations of the renewal function are also suggested.

SYMMETRIES OF COMPACT CONVEX SETS

1974 ◽  
Vol 25 (1) ◽  
pp. 323-328 ◽  
Author(s):  
E. B. DAVIES
Keyword(s):  
Convex Sets ◽  
Compact Convex
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