The extreme bodies in the set of plane convex bodies with a given width function

1975 ◽  
Vol 22 (3-4) ◽  
pp. 203-207 ◽  
Author(s):  
Michael Kallay
Keyword(s):  
Convex Bodies ◽  
Plane Convex ◽  
Width Function ◽  
Plane Convex Bodies

Decomposing plane convex bodies

1972 ◽  
Vol 23 (1) ◽  
pp. 534-536 ◽  
Author(s):  
Walter Meyer
Keyword(s):  
Convex Bodies ◽  
Plane Convex ◽  
Plane Convex Bodies

Step by step approximation of plane convex bodies

1992 ◽  
Vol 58 (6) ◽  
pp. 606-610 ◽  
Author(s):  
Josef S. M�ller
Keyword(s):  
Convex Bodies ◽  
Plane Convex ◽  
Step Approximation ◽  
Plane Convex Bodies

scholarly journals Upper bounds for the perimeter of plane convex bodies

2013 ◽  
Vol 142 (2) ◽  
pp. 366-383 ◽  
Author(s):  
Alexey Glazyrin ◽  
Filip Morić
Keyword(s):  
Convex Bodies ◽  
Upper Bounds ◽  
Plane Convex ◽  
Plane Convex Bodies

Geometric Study of Minkowski Differences of Plane Convex Bodies

2006 ◽  
Vol 58 (3) ◽  
pp. 600-624 ◽  
Author(s):  
Yves Martinez-Maure
Keyword(s):  
Convex Bodies ◽  
Minkowski Inequality ◽  
Continuous Functions ◽  
Support Functions ◽  
Plane Convex ◽  
Arbitrary Convex ◽  
The Difference ◽  
Minkowski Difference ◽  
Plane Convex Bodies ◽  
Geometric Study

AbstractIn the Euclidean plane ℝ2, we define the Minkowski difference 𝒦–𝓛 of two arbitrary convex bodies 𝒦, 𝓛 as a rectifiable closed curve ℋh⊂ ℝ2that is determined by the differenceh=h𝒦–h𝓛of their support functions. This curve ℋhis called the hedgehog with support functionh. More generally, the object of hedgehog theory is to study the Brunn–Minkowski theory in the vector space of Minkowski differences of arbitrary convex bodies of Euclidean space ℝn+1, defined as (possibly singular and self-intersecting) hypersurfaces of ℝn+1. Hedgehog theory is useful for: (i) studying convex bodies by splitting them into a sum in order to reveal their structure; (ii) converting analytical problems into geometrical ones by considering certain real functions as support functions. The purpose of this paper is to give a detailed study of plane hedgehogs, which constitute the basis of the theory. In particular: (i) we study their lengthmeasures and solve the extension of the Christoffel–Minkowski problemto plane hedgehogs; (ii) we characterize support functions of plane convex bodies among support functions of plane hedgehogs and support functions of plane hedgehogs among continuous functions; (iii) we study the mixed area of hedgehogs in ℝ2and give an extension of the classical Minkowski inequality (and thus of the isoperimetric inequality) to hedgehogs.

scholarly journals Geometric Inequalities for Plane Convex Bodies

1979 ◽  
Vol 22 (1) ◽  
pp. 9-16 ◽  
Author(s):  
G. D. Chakerian
Keyword(s):  
Euclidean Plane ◽  
Compact Convex ◽  
Convex Bodies ◽  
Plane Convex ◽  
Left Hand ◽  
Supporting Line ◽  
Plane Convex Body ◽  
The Right ◽  
Derivatives Of ◽  
Plane Convex Bodies

In what follows we shall mean by a plane convex body K a compact convex subset of the Euclidean plane having nonempty interior. We shall denote by h (K, θ) the supporting function of K restricted to the unit circle. This measures the signed distances from the origin to the supporting line of K with outward normal (cos θ, sin θ). The right hand and left hand derivatives of h (K, θ) exist everywhere and are equal except on a countable set.

scholarly journals Equilibria of Plane Convex Bodies

2021 ◽  
Vol 31 (5) ◽  
Author(s):  
Jonas Allemann ◽  
Norbert Hungerbühler ◽  
Micha Wasem
Keyword(s):  
Convex Body ◽  
Center Of Mass ◽  
Convex Bodies ◽  
Winding Number ◽  
Plane Convex ◽  
Planar Convex ◽  
Plane Convex Bodies

AbstractWe obtain a formula for the number of horizontal equilibria of a planar convex body K with respect to a center of mass O in terms of the winding number of the evolute of $$\partial K$$ ∂ K with respect to O. The formula extends to the case where O lies on the evolute of $$\partial K$$ ∂ K and a suitably modified version holds true for non-horizontal equilibria.

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