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.

The Measure of Plane Sets

1943 ◽  
Vol 39 (1) ◽  
pp. 51-53
Author(s):  
P. A. P. Moran
Keyword(s):  
Finite Number ◽  
Coordinate System ◽  
Lebesgue Measure ◽  
Upper Bound ◽  
Two Dimensional ◽  
Image Position ◽  
Bounded Sets ◽  
Dimensional Lebesgue Measure

We consider bounded sets in a plane. If X is such a set, we denote by Pθ(X) the projection of X on the line y = x tan θ, where x and y belong to some fixed coordinate system. By f(θ, X) we denote the measure of Pθ(X), taking this, in general, as an outer Lebesgue measure. The least upper bound of f (θ, X) for all θ we denote by M. We write sm X for the outer two-dimensional Lebesgue measure of X. Then G. Szekeres(1) has proved that if X consists of a finite number of continua,Béla v. Sz. Nagy(2) has obtained a stronger inequality, and it is the purpose of this paper to show that these results hold for more general classes of sets.

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.

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.

On Periodic Solutions of x‴ + ax″ + b′ + g(x) = 0

1967 ◽  
Vol 10 (1) ◽  
pp. 75-77 ◽  
Author(s):  
R.R.D. Kemp
Keyword(s):  
Periodic Solutions ◽  
Dimensional Space ◽  
Standard Theory ◽  
Two Dimensional ◽  
Image Position

In [1] J. O. C. Ezeilo asks whether the equation1has periodic solutions for a ≠ 0. Since (1) has a two-dimensional space of solutions of period 2π if sin x is approximated by x, it is plausible to conclude, by analogy with x″ + sin x = 0, that (1) does have periodic solutions. However, when one applies the standard theory of perturbation of periodic solutions (treating a as small, see [2]), one finds that the only real periodic solutions obtainable in this manner are the trivial ones x(t, a) = nπ for some integer n.

Cushions, cigars and diamonds: an area-perimeter problem for symmetric ovals

1979 ◽  
Vol 85 (1) ◽  
pp. 1-16 ◽  
Author(s):  
H. T. Croft
Keyword(s):  
Upper Bound ◽  
Natural Condition ◽  
Lattice Point ◽  
Convex Set ◽  
General Setting ◽  
Lattice Points ◽  
Two Dimensional ◽  
Algebraic Expressions ◽  
Image Position ◽  
Closed Convex Set

P. R. Scott (1) has asked which two-dimensional closed convex set E, centro-symmetric in the origin O, and containing no other Cartesian lattice-point in its interior, maximizes the ratio A/P, where A, P are the area, perimeter of E; he conjectured that the answer is the ‘rounded square’ (‘cushion’ in what follows), described below. We shall prove this, indeed in a more general setting, by seeking to maximizewhere κ is a parameter (0 < κ < 2); the set of admissible E is those E centro-symmetric in 0 that do not contain in their interior certain fixed lattice-points. There are two problems, the unrestricted one , where there is no given upper bound on A (it will become apparent that this problem only has a finite answer when κ ≥ 1) and the restricted one , when one is given a bound B and we must have A ≤ B. Special interest attaches to the case B = 4, both because of Minkowski's theorem: any E symmetric in O and containing no other lattice-point has area at most 4; and because it turns out that it is a ‘natural’ condition: the algebraic expressions simplify to a remarkable extent. Hence in what follows, the ‘restricted case ’ shall mean A ≤ 4.

3-D Graphing of XRF Matrix Correction Equations

1994 ◽  
Vol 38 ◽  
pp. 649-656
Author(s):  
Anthony J. Klimasara
Keyword(s):  
Dimensional Space ◽  
Correction Term ◽  
Three Dimensional ◽  
Calibration Plot ◽  
Two Dimensional ◽  
Matrix Correction ◽  
Image Position ◽  
Correction Equations ◽  
Calibration Plane ◽  
Three Dimensional Space

Abstract The Lachance-Traill, and Lucas-Tooth-Price matrix correction equations/functions for XRF determined concentrations can be graphically interpreted with the help of three dimensional graphics. Statistically derived Lachance-Traill and Lucas-Tooth-Price matrix correction equations can be represented as follows: 1 where: Ci -elemental concentration of element “i” Ij -X-Ray intensity representing element “i” Ai0 -regression intercept for element “i” Ai -regression coefficient Zj -correction term defined below 2 Ai0, Aj , and Zi together represent the results of a multi-dimensional contribution. li, Ci, and Zi can be represented in three dimensional Cartesian space by X, Y and Z. These three variables are connected by a matrix correction equation that can be graphed as the function Y = F(X, Z), which represents a plane in three dimensional space. It can be seen that each chemical element will deliver a different set of coefficients in the equation of a plane that is called here a calibration plane. The commonly known and used two dimensional calibration plot is a “shadow” of the three dimensional real calibration points. These real (not shadow) points reside on a regression calibration plane in this three dimensional space. Lachance-Traill and Lucas-Tooth-Price matrix correction equations introduce the additional dimension(s) to the two dimensional flat image of uncorrected data. Illustrative examples generated by three dimensional graphics will be presented.

On Plane Sections and Projections of Convex Sets

1969 ◽  
Vol 21 ◽  
pp. 1331-1337
Author(s):  
H. Groemer
Keyword(s):  
Convex Body ◽  
Orthogonal Projection ◽  
Convex Sets ◽  
Convex Set ◽  
Dimensional Space ◽  
Compact Convex ◽  
Three Dimensional ◽  
Plane Section ◽  
Compact Convex Set ◽  
Three Dimensional Space

Let K be a three-dimensional convex body. It has been conjectured (cf. 3) that one can always find a plane H such that the intersection K ∩ H is, in a certain sense, fairly circular. Instead of the plane section K ∩ H one can also consider the orthogonal projection of K onto H. Our aim in this paper is to prove some results concerning this type of problems. It appears that John has found similar theorems (cf. the remarks of Behrend, 1, p. 717). His proof of the first inequality of our Theorem 1 has been published (6). It is based on a property of the ellipse of inertia which will not be used in the present paper.A non-empty compact convex set 5 which is contained in some plane of euclidean three-dimensional space E3 will be called a convex domain.

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