Equilibria of 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.

Download Full-text

Decomposing plane convex bodies

Archiv der Mathematik ◽

10.1007/bf01304929 ◽

1972 ◽

Vol 23 (1) ◽

pp. 534-536 ◽

Cited By ~ 2

Author(s):

Walter Meyer

Keyword(s):

Convex Bodies ◽

Plane Convex ◽

Plane Convex Bodies

Download Full-text

Retrieving convex bodies from restricted covariogram functions

Advances in Applied Probability ◽

10.1239/aap/1189518630 ◽

2007 ◽

Vol 39 (3) ◽

pp. 613-629 ◽

Cited By ~ 6

Author(s):

Gennadiy Averkov ◽

Gabriele Bianchi

Keyword(s):

Convex Body ◽

Lebesgue Measure ◽

Dimensional Space ◽

Convex Bodies ◽

Baire Category ◽

The Body ◽

Planar Case ◽

Large Classes ◽

Negative Results ◽

Planar Convex

The covariogram gK(x) of a convex body K ⊆ Ed is the function which associates to each x ∈ Ed the volume of the intersection of K with K + x, where Ed denotes the Euclidean d-dimensional space. Matheron (1986) asked whether gK determines K, up to translations and reflections in a point. Positive answers to Matheron's question have been obtained for large classes of planar convex bodies, while for d ≥ 3 there are both positive and negative results. One of the purposes of this paper is to sharpen some of the known results on Matheron's conjecture indicating how much of the covariogram information is needed to get the uniqueness of determination. We indicate some subsets of the support of the covariogram, with arbitrarily small Lebesgue measure, such that the covariogram, restricted to those subsets, identifies certain geometric properties of the body. These results are more precise in the planar case, but some of them, both positive and negative ones, are proved for bodies of any dimension. Moreover some results regard most convex bodies, in the Baire category sense. Another purpose is to extend the class of convex bodies for which Matheron's conjecture is confirmed by including all planar convex bodies possessing two nondegenerate boundary arcs being reflections of each other.

Download Full-text

Cheeger Sets for Rotationally Symmetric Planar Convex Bodies

Results in Mathematics ◽

10.1007/s00025-021-01539-7 ◽

2021 ◽

Vol 77 (1) ◽

Author(s):

Antonio Cañete

Keyword(s):

Convex Body ◽

Convex Bodies ◽

Rotationally Symmetric ◽

Planar Convex ◽

Cheeger Sets

AbstractIn this note we obtain some properties of the Cheeger set $$C_\varOmega $$ C Ω associated to a k-rotationally symmetric planar convex body $$\varOmega $$ Ω . More precisely, we prove that $$C_\varOmega $$ C Ω is also k-rotationally symmetric and that the boundary of $$C_\varOmega $$ C Ω touches all the edges of $$\varOmega $$ Ω .

Download Full-text