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.

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.

scholarly journals On the relative distances of seven points in a plane convex body

2007 ◽  
Vol 87 (1-2) ◽  
pp. 83-95 ◽  
Author(s):  
Antal Joós ◽  
Zsolt Lángi
Keyword(s):  
Convex Body ◽  
Plane Convex ◽  
Plane Convex Body

scholarly journals On the relative distances of six points in a plane convex body

2005 ◽  
Vol 42 (3) ◽  
pp. 253-264
Author(s):  
Károly Böröczky ◽  
Zsolt Lángi
Keyword(s):  
Convex Body ◽  
Upper Bound ◽  
Euclidean Distance ◽  
Euclidean Plane ◽  
Relative Distance ◽  
Plane Convex ◽  
Euclidean Length ◽  
Plane Convex Body

Let C be a convex body in the Euclidean plane. By the relative distance of points p and q we mean the ratio of the Euclidean distance of p and q to the half of the Euclidean length of a longest chord of C parallel to pq. In this note we find the least upper bound of the minimum pairwise relative distance of six points in a plane convex body.

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.

scholarly journals Bounding Regions to Plane Steepest Descent Curves of Quasiconvex Families

2016 ◽  
Vol 2016 ◽  
pp. 1-17
Author(s):  
Marco Longinetti ◽  
Paolo Manselli ◽  
Adriana Venturi
Keyword(s):  
Convex Body ◽  
Support Function ◽  
Sufficient Conditions ◽  
Minimal Length ◽  
Steepest Descent ◽  
Opposite Orientation ◽  
Plane Convex ◽  
Plane Convex Body ◽  
Arbitrary Plane ◽  
Constructed Self

Two-dimensional steepest descent curves (SDC) for a quasiconvex family are considered; the problem of their extensions (with constraints) outside of a convex bodyKis studied. It is shown that possible extensions are constrained to lie inside of suitable bounding regions depending onK. These regions are bounded by arcs of involutes of∂Kand satisfy many inclusions properties. The involutes of the boundary of an arbitrary plane convex body are defined and written by their support function. Extensions SDC of minimal length are constructed. Self-contracting sets (with opposite orientation) are considered: necessary and/or sufficient conditions for them to be subsets of SDC are proved.

Sign in / Sign up

Export Citation Format

Share Document