Order types are a well known abstraction of combinatorial properties of a point set. By Mn "ev's universality theorem for each semi-algebraic set @math there is an order type with a realization space that is to @math . We consider realization spaces of , a refinement of order types. We show that the realization spaces of allowable sequences are and consequently deciding the realizability is complete in the ( ). This result holds even if the realization space of the order type induced by the allowable sequence is non-empty. Furthermore, we argue that our result is a useful tool for further geometric reductions. We support this by giving -hardness proofs for the realizability of abstract convex geometries and for the recognition problem of visibility graphs of polygons with holes using the hardness result for allowable sequences. This solves two longstanding open problems.
In their paper Edelman and Jamison @cite_11 have developed the foundations of a combinatorial abstraction of convexity. Similar ideas were studied by Dilworth @cite_31 and later by Korte, Lov 'a sz and Schrader @cite_42 via the notion of , a concept dual to the one of a finite convex geometry. Today, the concept of convex geometry appears in many fields of mathematics such as formal language theory @cite_45 , choice theory @cite_20 and mathematical psychology @cite_17 among others. @cite_14 showed that any abstract convex geometry can be represented as generalized shelling'' in @math for some @math . Richters and Rogers @cite_5 reproved this theorem giving a better bound on the dimension. Different representations of finite convex geometries using different shapes than points for the ground set have been studied @cite_18 @cite_38 @cite_6 .
@cite_38 @cite_18 @cite_14 @cite_42 @cite_17 @cite_6 @cite_45 @cite_5 @cite_31 @cite_20 @cite_11
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The notion of an abstract convex geometry offers an abstraction of the standard notion of convexity in a linear space. Kashiwabara, Nakamura and Okamoto introduce the notion of a generalized convex shelling into @math and prove that a convex geometry may always be represented with such a shelling. We provide a new, shorter proof of their result using a recent representation theorem of Richter and Rubinstein, and deduce a different upper bound on the dimension of the shelling. Abstract Let F be a finite set of circles in the plane. The usual convex closure restricted to F yields a convex geometry, which is a combinatorial structure introduced by P. H. Edelman in 1980 under the name “anti-exchange closure system”. We prove that if the circles are collinear and they are arranged in a “concave way”, then they determine a convex geometry of convex dimension at most 2, and each finite convex geometry of convex dimension at most 2 can be represented this way. The proof uses some recent results from lattice theory, and some of the auxiliary statements on lattices or convex geometries could be of separate interest. The paper concludes with some open problems. Finite convex geometries are combinatorial structures. It follows from a recent result of M. Richter and L.G. Rogers that there is an infinite set @math of planar convex polygons such that @math with respect to geometric convex hulls is a locally convex geometry and every finite convex geometry can be represented by restricting the structure of @math to a finite subset in a natural way. An @math is a differentiable convex simple closed curve @math in the plane having an inscribed circle of radius @math and a circumscribed circle of radius @math such that the ratio @math is at least @math . Motivated by Richter and Rogers' result, we construct a set @math such that (1) @math contains all points of the plane as degenerate singleton circles and all of its non-singleton members are differentiable convex simple closed planar curves; (2) @math with respect to the geometric convex hull operator is a locally convex geometry; (3) as opposed to @math , @math is closed with respect to non-degenerate affine transformations; and (4) for every (small) positive @math and for every finite convex geometry, there are continuum many pairwise affine-disjoint finite subsets @math of @math such that each @math consists of almost-circles of accuracy @math and the convex geometry in question is represented by restricting the convex hull operator to @math . The affine-disjointness of @math and @math means that, in addition to @math , even @math is disjoint from @math for every non-degenerate affine transformation @math . In this paper we present several results on the expected complexity of a convex hull of @math points chosen uniformly and independently from a convex shape. (i) We show that the expected number of vertices of the convex hull of @math points, chosen uniformly and independently from a disk is @math , and @math for the case a convex polygon with @math sides. Those results are well known (see rs-udkhv-63,r-slcdn-70,ps-cgi-85 ), but we believe that the elementary proof given here are simpler and more intuitive. (ii) Let @math be a set of directions in the plane, we define a generalized notion of convexity induced by @math , which extends both rectilinear convexity and standard convexity. We prove that the expected complexity of the @math -convex hull of a set of @math points, chosen uniformly and independently from a disk, is @math , where @math is the largest angle between two consecutive vectors in @math . This result extends the known bounds for the cases of rectilinear and standard convexity. (iii) Let @math be an axis parallel hypercube in @math . We prove that the expected number of points on the boundary of the quadrant hull of a set @math of @math points, chosen uniformly and independently from @math is @math . Quadrant hull of a set of points is an extension of rectilinear convexity to higher dimensions. In particular, this number is larger than the number of maxima in @math , and is also larger than the number of points of @math that are vertices of the convex hull of @math . Those bounds are known bkst-anmsv-78 , but we believe the new proof is simpler.