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.

scholarly journals On the relative lengths of sides of convex polygons

2003 ◽  
Vol 40 (1-2) ◽  
pp. 115-120
Author(s):  
Zs. Lángi
Keyword(s):  
Convex Body ◽  
Euclidean Distance ◽  
Upper Bounds ◽  
Relative Distance ◽  
Convex Polygons ◽  
Euclidean Length

Let C be a convex body. 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. The aim of the paper is to find upper bounds for the minimum of the relative lengths of the sides of convex hexagons and heptagons.

scholarly journals Steinhardt's inequality in the Minkowski plane

1992 ◽  
Vol 45 (2) ◽  
pp. 261-266 ◽  
Author(s):  
Mostafa Ghandehari
Keyword(s):  
Convex Body ◽  
Unit Circle ◽  
Minkowski Plane ◽  
Plane Convex ◽  
Euclidean Length ◽  
Plane Convex Body

In a Minkowski plane with unit circle E, the product of the positive circumference of a plane convex body K and that of its polar dual is greater than or equal to the square of the Euclidean length of the polar dual of E. Equality holds if and only if K is a Euclidean unit circle.

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 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.

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