scholarly journals Retrieving convex bodies from restricted covariogram functions

2007 ◽  
Vol 39 (3) ◽  
pp. 613-629 ◽  
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.

scholarly journals Retrieving convex bodies from restricted covariogram functions

2007 ◽  
Vol 39 (03) ◽  
pp. 613-629 ◽  
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 covariogramgK(x) of a convex bodyK⊆Edis the function which associates to eachx∈Edthe volume of the intersection ofKwithK+x, whereEddenotes the Euclideand-dimensional space. Matheron (1986) asked whethergKdeterminesK, 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 ford≥ 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.

scholarly journals Centrally symmetric convex bodies and sections having maximal quermassintegrals

2012 ◽  
Vol 49 (2) ◽  
pp. 189-199
Author(s):  
E. Makai ◽  
H. Martini
Keyword(s):  
Convex Body ◽  
Convex Bodies ◽  
The Body ◽  
Symmetric Convex ◽  
Local Theorem ◽  
Centrally Symmetric ◽  
Euclidean Unit Ball ◽  
Analogous Theorem ◽  
Lower Dimensional ◽  
Dimensional Surface

Let d ≧ 2, and let K ⊂ ℝd be a convex body containing the origin 0 in its interior. In a previous paper we have proved the following. The body K is 0-symmetric if and only if the following holds. For each ω ∈ Sd−1, we have that the (d − 1)-volume of the intersection of K and an arbitrary hyperplane, with normal ω, attains its maximum if the hyperplane contains 0. An analogous theorem, for 1-dimensional sections and 1-volumes, has been proved long ago by Hammer (see [2]). In this paper we deal with the ((d − 2)-dimensional) surface area, or with lower dimensional quermassintegrals of these intersections, and prove an analogous, but local theorem, for small C2-perturbations, or C3-perturbations of the Euclidean unit ball, respectively.

scholarly journals Cheeger Sets for Rotationally Symmetric Planar Convex Bodies

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 .

Generalized Convex Bodies of Revolution

1967 ◽  
Vol 19 ◽  
pp. 972-996 ◽  
Author(s):  
WM. J. Firey
Keyword(s):  
Convex Body ◽  
Dimensional Space ◽  
Convex Bodies ◽  
Three Dimensional ◽  
Body Of Revolution ◽  
Bodies Of Revolution ◽  
Cartesian Coordinates ◽  
Three Dimensional Space

The figures studied in this paper are special convex bodies in Euclidean three-dimensional space which we shall call generalized convex bodies of revolution (GCBR). Such a set is obtained by the following procedure. Let K1 be a convex body of revolution and let x, y, z denote Cartesian coordinates in a system for which the z-axis is the axis of K1.

scholarly journals On the volume product of planar polar convex bodies — Lower estimates with stability

2013 ◽  
Vol 50 (2) ◽  
pp. 159-198
Author(s):  
K. Böröczky ◽  
E. Makai ◽  
M. Meyer ◽  
S. Reisner
Keyword(s):  
Convex Body ◽  
Rotational Symmetry ◽  
Polar Body ◽  
Convex Bodies ◽  
Stability Estimate ◽  
The Body ◽  
Symmetric Convex Body ◽  
Symmetric Convex ◽  
The Stability ◽  
Area Product

Let K ⊂ ℝ2 be an o-symmetric convex body, and K* its polar body. Then we have |K| · |K*| ≧ 8, with equality if and only if K is a parallelogram. (|·| denotes volume). If K ⊂ ℝ2 is a convex body, with o ∈ int K, then |K| · |K*| ≧ 27/4, with equality if and only if K is a triangle and o is its centroid. If K ⊂ ℝ2 is a convex body, then we have |K| · |[(K − K)/2)]*| ≧ 6, with equality if and only if K is a triangle. These theorems are due to Mahler and Reisner, Mahler and Meyer, and to Eggleston, respectively. We show an analogous theorem: if K has n-fold rotational symmetry about o, then |K| · |K*| ≧ n2 sin2(π/n), with equality if and only if K is a regular n-gon of centre o. We will also give stability variants of these four inequalities, both for the body, and for the centre of polarity. For this we use the Banach-Mazur distance (from parallelograms, or triangles), or its analogue with similar copies rather than affine transforms (from regular n-gons), respectively. The stability variants are sharp, up to constant factors. We extend the inequality |K| · |K*| ≧ n2 sin2(π/n) to bodies with o ∈ int K, which contain, and are contained in, two regular n-gons, the vertices of the contained n-gon being incident to the sides of the containing n-gon. Our key lemma is a stability estimate for the area product of two sectors of convex bodies polar to each other. To several of our statements we give several proofs; in particular, we give a new proof for the theorem of Mahler-Reisner.

scholarly journals An Upper Bound for Volumes of Convex Bodies

1969 ◽  
Vol 9 (3-4) ◽  
pp. 503-510
Author(s):  
William J. Firey
Keyword(s):  
Convex Body ◽  
Orthogonal Projection ◽  
Upper Bound ◽  
Linear Subspace ◽  
Dimensional Space ◽  
Convex Bodies ◽  
Maximum Length ◽  
Two Dimensional ◽  
Image Position

Consider a non-degenerate convex body K in a Euclidean (n + 1)-dimensional space of points (x, z) = (x1,…, xn, z) where n ≧2. Denote by μ the maximum length of segments in K which are parallel to the z-axis, and let Aj, signify the area (two dimensional volume) of the orthogonal projection of K onto the linear subspace spanned by the z- and xj,-axes. We shall prove that the volume V(K) of K satisfies After this, some applications of (1) are discussed.

scholarly journals Inradius and circumradius for planar convex bodies containing no lattice points

1999 ◽  
Vol 59 (1) ◽  
pp. 163-168
Author(s):  
P.R. Scott ◽  
P.W. Awyong
Keyword(s):  
Convex Body ◽  
Convex Bodies ◽  
Integer Lattice ◽  
Lattice Points ◽  
Planar Convex

Let K be a planar convex body containing no points of the integer lattice. We give a new inequality relating the inradius and circumradius of K.

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.

scholarly journals Inequalities for the Surface Area of Projections of Convex Bodies

2018 ◽  
Vol 70 (4) ◽  
pp. 804-823 ◽  
Author(s):  
Apostolos Giannopoulos ◽  
Alexander Koldobsky ◽  
Petros Valettas
Keyword(s):  
Convex Body ◽  
Surface Area ◽  
Convex Bodies ◽  
Maximal Surface ◽  
Projection Body ◽  
Projections Of Convex Bodies ◽  
Lower Dimensional

AbstractWe provide general inequalities that compare the surface area S(K) of a convex body K in ℝn to the minimal, average, or maximal surface area of its hyperplane or lower dimensional projections. We discuss the same questions for all the quermassintegrals of K. We examine separately the dependence of the constants on the dimension in the case where K is in some of the classical positions or K is a projection body. Our results are in the spirit of the hyperplane problem, with sections replaced by projections and volume by surface area.

Prescribing Centro-Affine Curvature From One Convex Body to Another

2020 ◽  
Author(s):  
Alina Stancu
Keyword(s):  
Convex Body ◽  
Convex Bodies ◽  
Minkowski Inequality ◽  
Curvature Flow ◽  
Constant Volume ◽  
Convex Hypersurface ◽  
Strictly Convex ◽  
Initial Hypersurface ◽  
Affine Curvature ◽  
Convex Hypersurfaces

Abstract We study a curvature flow on smooth, closed, strictly convex hypersurfaces in $\mathbb{R}^n$, which commutes with the action of $SL(n)$. The flow shrinks the initial hypersurface to a point that, if rescaled to enclose a domain of constant volume, is a smooth, closed, strictly convex hypersurface in $\mathbb{R}^n$ with centro-affine curvature proportional, but not always equal, to the centro-affine curvature of a fixed hypersurface. We outline some consequences of this result for the geometry of convex bodies and the logarithmic Minkowski inequality.

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