scholarly journals Asymmetry of a plane convex set with respect to its centroid

1958 ◽  
Vol 8 (2) ◽  
pp. 335-337 ◽  
Author(s):  
B. M. Stewart
Keyword(s):  
Convex Set ◽  
Plane Convex

Maximal Areas of Reuleaux Polygons

1970 ◽  
Vol 13 (2) ◽  
pp. 175-179 ◽  
Author(s):  
G. T. Sallee
Keyword(s):  
Finite Number ◽  
Convex Set ◽  
Constant Width ◽  
Plane Convex ◽  
Circular Arcs ◽  
Positive Length ◽  
Reuleaux Polygon ◽  
Reuleaux Polygons

In this paper we provide new proofs of some interesting results of Firey [2] on isoperimetric ratios of Reuleaux polygons. Recall that a Reuleaux polygon is a plane convex set of constant width whose boundary consists of a finite (odd) number of circular arcs. Equivalently, it is the intersection of a finite number of suitably chosen congruent discs. For more details, see [1, p. 128].If a Reuleaux polygon has n sides (arcs) of positive length (where n is odd and ≥ 3), we will refer to it as a Reuleaux n-gon, or sometimes just as an n-gon. If all of the sides are equal, it is termed a regular n-gon.

Applications of outwardly simple line families to plane convex sets

1977 ◽  
Vol 82 (3) ◽  
pp. 353-356 ◽  
Author(s):  
K. J. Falconer
Keyword(s):  
Convex Sets ◽  
Convex Set ◽  
Plane Convex ◽  
Minimal Width ◽  
Parallel Lines

We use the concept of outwardly simple line families (see Hammer and Sobczyk (4)) first to obtain conditions involving mid-chords that ensure that a plane convex set is centro-symmetric, and secondly to show that it is possible to inscribe a semicircle of diameter ω in any convex set of minimal width ω in at least 3 different ways. We show that a plane convex set X is centro-symmetric if every mid-chord of X (that is every chord of X mid-way between two parallel lines of support of X) bisects the area of X, or alternatively if every mid-chord of X is a diameter of X (that is a longest chord in some direction). Hammer and Smith (3) have used outwardly simple line families in a different way to show that a plane convex set X is centro-symmetric if every diameter of X bisects the area of X, or if every diameter of X bisects the perimeter of X.

scholarly journals Diameter, width and thickness in the hyperbolic plane

2021 ◽  
Vol 112 (3) ◽  
Author(s):  
Ákos G. Horváth
Keyword(s):  
Convex Body ◽  
Hyperbolic Geometry ◽  
Hyperbolic Plane ◽  
Convex Set ◽  
Constant Width ◽  
Plane Convex ◽  
Plane Convex Body ◽  
Constant Diameter

AbstractIn hyperbolic geometry there are several concepts to measure the breadth or width of a convex set. In the first part of the paper we collect them and compare their properties. Than we introduce a new concept to measure the width and thickness of a convex body. Correspondingly, we define three classes of bodies, bodies of constant with, bodies of constant diameter and bodies having the constant shadow property, respectively. We prove that the property of constant diameter follows to the fulfilment of constant shadow property, and both of them are stronger as the property of constant width. In the last part of this paper, we introduce the thickness of a constant body and prove a variant of Blaschke’s theorem on the larger circle inscribed to a plane-convex body of given thickness and diameter.

Sharpening an Estimate of the Size of the Sumset of a Convex Set

2020 ◽  
Vol 107 (5-6) ◽  
pp. 984-987
Author(s):  
K. I. Ol’mezov
Keyword(s):  
Convex Set

scholarly journals Horosphere slab separation theorems in manifolds without conjugate points

2019 ◽  
Vol 27 (1) ◽  
Author(s):  
Sameh Shenawy
Keyword(s):  
Euclidean Space ◽  
Hyperbolic Space ◽  
Existence And Uniqueness ◽  
Convex Set ◽  
Sufficient Conditions ◽  
Conjugate Points ◽  
Separation Theorems ◽  
Farthest Points ◽  
Simply Connected ◽  
Convex Subsets

Abstract Let $\mathcal {W}^{n}$ W n be the set of smooth complete simply connected n-dimensional manifolds without conjugate points. The Euclidean space and the hyperbolic space are examples of these manifolds. Let $W\in \mathcal {W}^{n}$ W ∈ W n and let A and B be two convex subsets of W. This note aims to investigate separation and slab horosphere separation of A and B. For example,sufficient conditions on A and B to be separated by a slab of horospheres are obtained. Existence and uniqueness of foot points and farthest points of a convex set A in $W\in \mathcal {W}$ W ∈ W are considered.

An extremal property of the hypersphere

1951 ◽  
Vol 47 (1) ◽  
pp. 245-247 ◽  
Author(s):  
A. M. Macbeath
Keyword(s):  
Convex Body ◽  
Extremal Property ◽  
Plane Case ◽  
Simple Application ◽  
Plane Convex ◽  
Plane Convex Body

It was shown by Sas (1) that, if K is a plane convex body, then it is possible to inscribe in K a convex n-gon occupying no less a fraction of its area than the regular n-gon occupies in its circumscribing circle. It is the object of this note to establish the n-dimensional analogue of Sas's result, giving incidentally an independent proof of the plane case. The proof is a simple application of the Steiner method of symmetrization.

On random divisions of a convex set

1978 ◽  
Vol 15 (3) ◽  
pp. 645-649 ◽  
Author(s):  
Svante Janson
Keyword(s):  
Elementary Proof ◽  
Convex Set ◽  
Limiting Distribution ◽  
Independent Identically Distributed ◽  
Normally Distributed

This paper gives an elementary proof that, under some general assumptions, the number of parts a convex set in Rd is divided into by a set of independent identically distributed hyperplanes is asymptotically normally distributed. An example is given where the distribution of hyperplanes is ‘too singular' to satisfy the assumptions, and where a different limiting distribution appears.

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