Two Inequalities for Planar Convex Sets

1985 ◽  
Vol 28 (1) ◽  
pp. 60-66 ◽  
Author(s):  
George Tsintsifas
Keyword(s):  
Convex Sets ◽  
Convex Set ◽  
Compact Convex ◽  
Compact Convex Set ◽  
Planar Convex ◽  
Straight Lines

AbstractB. Grünbaum, J. N. Lillington and lately R. J. Gardner, S. Kwapien and D. P. Laurie have considered inequalities defined by three concurrent straight lines in the interior of a planar compact convex set. In this note we prove two elegant conjectures by R. J. Gardner, S. Kwapien and D. P. Laurie.

scholarly journals Circumradius-diameter and width-inradius relations for lattice constrained convex sets

1999 ◽  
Vol 59 (1) ◽  
pp. 147-152 ◽  
Author(s):  
Poh Wah Awyong ◽  
Paul R. Scott
Keyword(s):  
Lattice Point ◽  
Convex Sets ◽  
Convex Set ◽  
Compact Convex ◽  
Integer Lattice ◽  
Lattice Points ◽  
Compact Convex Set ◽  
Planar Convex ◽  
Rectangular Lattices

Let K be a planar, compact, convex set with circumradius R, diameter d, width w and inradius r, and containing no points of the integer lattice. We generalise inequalities concerning the ‘dual’ quantities (2R − d) and (w − 2r) to rectangular lattices. We then use these results to obtain corresponding inequalities for a planar convex set with two interior lattice points. Finally, we conjecture corresponding results for sets containing one interior lattice point.

Rates of convergence for random approximations of convex sets

1996 ◽  
Vol 28 (02) ◽  
pp. 384-393 ◽  
Author(s):  
Lutz Dümbgen ◽  
Günther Walther
Keyword(s):  
Convex Subset ◽  
Hausdorff Distance ◽  
Convex Sets ◽  
Convex Set ◽  
Compact Convex ◽  
Almost Sure Convergence ◽  
Rates Of Convergence ◽  
Random Sets ◽  
Compact Convex Set

The Hausdorff distance between a compact convex set K ⊂ ℝd and random sets is studied. Basic inequalities are derived for the case of being a convex subset of K. If applied to special sequences of such random sets, these inequalities yield rates of almost sure convergence. With the help of duality considerations these results are extended to the case of being the intersection of a random family of halfspaces containing K.

Exact distributions for shapes of random triangles in convex sets

1985 ◽  
Vol 17 (02) ◽  
pp. 308-329 ◽  
Author(s):  
D. G. Kendall
Keyword(s):  
Convex Sets ◽  
Convex Set ◽  
Compact Convex ◽  
Direct Proof ◽  
Circular Disk ◽  
Compact Convex Set ◽  
Exact Evaluation ◽  
Exact Distributions ◽  
New Formula ◽  
Relevant Section

The paper starts with a simple direct proof that . A new formula is given for the shape-density for a triangle whose vertices are i.i.d.-uniform in a compact convex set K, and an exact evaluation of that shape-density is obtained when K is a circular disk. An (x, y)-diagram for an auxiliary shape-density is then introduced. When K = circular disk, it is shown that is virtually constant over a substantial region adjacent to the relevant section of the collinearity locus, large enough to contain the work-space for most collinearity studies, and particularly appropriate when the ‘strip’ method is used to assess near-collinearity.

scholarly journals On the maximal circumradius of a planar convex set containing one lattice point

1995 ◽  
Vol 52 (1) ◽  
pp. 137-151 ◽  
Author(s):  
Poh W. Awyong ◽  
Paul R. Scott
Keyword(s):  
Lattice Point ◽  
Convex Set ◽  
Compact Convex ◽  
Lattice Points ◽  
Compact Convex Set ◽  
Planar Convex

We obtain a result about the maximal circumradius of a planar compact convex set having circumcentre O and containing no non-zero lattice points in its interior. In addition, we show that under certain conditions, the set with maximal circumradius is a triangle with an edge containing two lattice points.

scholarly journals Mazur's intersection property of balls for compact convex sets

1987 ◽  
Vol 35 (2) ◽  
pp. 267-274 ◽  
Author(s):  
J. H. M. Whitfield ◽  
V. Zizler
Keyword(s):  
Banach Space ◽  
Uniform Convergence ◽  
Convex Sets ◽  
Convex Set ◽  
Compact Convex ◽  
Extreme Points ◽  
Intersection Property ◽  
Compact Sets ◽  
Compact Convex Set ◽  
Mazur's Intersection Property

We show that every compact convex set in a Banach space X is an intersection of balls provided the cone generated by the set of all extreme points of the dual unit ball of X* is dense in X* in the topology of uniform convergence on compact sets in X. This allows us to renorm every Banach space with transfinite Schauder basis by a norm which shares the mentioned intersection property.

Exact distributions for shapes of random triangles in convex sets

1985 ◽  
Vol 17 (2) ◽  
pp. 308-329 ◽  
Author(s):  
D. G. Kendall
Keyword(s):  
Convex Sets ◽  
Convex Set ◽  
Compact Convex ◽  
Direct Proof ◽  
Circular Disk ◽  
Compact Convex Set ◽  
Exact Evaluation ◽  
Exact Distributions ◽  
New Formula ◽  
Relevant Section

The paper starts with a simple direct proof that . A new formula is given for the shape-density for a triangle whose vertices are i.i.d.-uniform in a compact convex set K, and an exact evaluation of that shape-density is obtained when K is a circular disk. An (x, y)-diagram for an auxiliary shape-density is then introduced. When K = circular disk, it is shown that is virtually constant over a substantial region adjacent to the relevant section of the collinearity locus, large enough to contain the work-space for most collinearity studies, and particularly appropriate when the ‘strip’ method is used to assess near-collinearity.

A geometric characterization concerning compact, convex sets

1991 ◽  
Vol 109 (2) ◽  
pp. 351-361 ◽  
Author(s):  
Christopher J. Mulvey ◽  
Joan Wick Pelletier
Keyword(s):  
Normed Space ◽  
Convex Sets ◽  
Convex Set ◽  
Compact Convex ◽  
Necessary And Sufficient Condition ◽  
Categorical Logic ◽  
Compact Convex Set ◽  
Necessary And Sufficient ◽  
Formal Statement

In this paper, we are concerned with establishing a characterization of any compact, convex set K in a normed space A in an arbitrary topos with natural number object. The characterization is geometric, not in the sense of categorical logic, but in the intuitive one, of describing any compact, convex set K in terms of simpler sets in the normed space A. It is a characterization of the compact, convex set in the sense that it provides a necessary and sufficient condition for an element of the normed space to lie within it. Having said this, we should immediately qualify our statement by stressing that this is the intuitive content of what is proved; the formal statement of the characterization is required to be in terms appropriate to the constructive context of the techniques used.

Weakly prime compact convex sets and uniform algebras

1977 ◽  
Vol 81 (2) ◽  
pp. 225-232 ◽  
Author(s):  
A. J. Ellis
Keyword(s):  
Convex Sets ◽  
Convex Set ◽  
Compact Convex ◽  
Uniform Algebra ◽  
Reduction Theory ◽  
Natural Analogue ◽  
Uniform Algebras ◽  
Compact Convex Set ◽  
Analytic Sets

1. Introduction. We introduce the notion of a weakly prime compact convex set, and we develop a reduction theory for spaces A(K). The notion is less restrictive in general than the prime compact convex sets of Chu, but gives a finer reduction than the Bishop and Silov decompositions forA(K) (12). The natural analogue for uniform algebras is related to the concept of weakly analytic sets due to Arenson, but unlike maximal weakly analytic sets the maximal weakly prime sets are always generalized peak sets; the uniform algebra can always be retrieved from the restrictions of the algebra to the maximal weakly prime sets.

Un théorème d'unicité pour les hyperplans poissoniens

1974 ◽  
Vol 11 (01) ◽  
pp. 184-189 ◽  
Author(s):  
G. Matheron
Keyword(s):  
Poisson Point Process ◽  
Convex Set ◽  
Compact Convex ◽  
Minkowski Sum ◽  
Line Segments ◽  
Supporting Function ◽  
Compact Convex Set ◽  
Finite Sums ◽  
Tangential Cone ◽  
Straight Lines

A stationary Poisson process of hyperplanes in Rn is characterized (up to an equivalence) by the function θ such that θ(s) is the density of the Poisson point process induced on the straight lines with direction s. The set of these functions θ is a convex cone ℛ1, a basis of which is a simplex Θ, and a given function θ belongs to ℛ1 if and only if it is the supporting function of a symmetrical compact convex set which is a finite Minkowski sum of line segments or the limit of such finite sums. Another application is given concerning the tangential cone at h = 0 of a coveriance function.

Un théorème d'unicité pour les hyperplans poissoniens

1974 ◽  
Vol 11 (1) ◽  
pp. 184-189 ◽  
Author(s):  
G. Matheron
Keyword(s):  
Poisson Point Process ◽  
Convex Set ◽  
Compact Convex ◽  
Minkowski Sum ◽  
Line Segments ◽  
Supporting Function ◽  
Compact Convex Set ◽  
Finite Sums ◽  
Tangential Cone ◽  
Straight Lines

A stationary Poisson process of hyperplanes in Rn is characterized (up to an equivalence) by the function θ such that θ(s) is the density of the Poisson point process induced on the straight lines with direction s. The set of these functions θ is a convex cone ℛ1, a basis of which is a simplex Θ, and a given function θ belongs to ℛ1 if and only if it is the supporting function of a symmetrical compact convex set which is a finite Minkowski sum of line segments or the limit of such finite sums. Another application is given concerning the tangential cone at h = 0 of a coveriance function.

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