scholarly journals On a Measure of Asymmetry of Convex Bodies

1962 ◽  
Vol 58 (2) ◽  
pp. 217-220 ◽  
Author(s):  
E. Asplund ◽  
E. Grosswald ◽  
B. Grünbaum
Keyword(s):  
Convex Body ◽  
Euclidean Space ◽  
Present Note ◽  
Convex Bodies ◽  
Dimensional Euclidean Space ◽  
Measure Of Asymmetry ◽  
Measures Of Asymmetry

In the present note we discuss some properties of a ‘measure of asymmetry’ of convex bodies in n-dimensional Euclidean space. Various measures of asymmetry have been treated in the literature (see, for example, (1), (6); references to most of the relevant results may be found in (4)). The measure introduced here has the somewhat surprising property that for n ≥ 3 the n-simplex is not the most asymmetric convex body in En. It seems to be the only measure of asymmetry for which this fact is known.

A Note on Covering by Convex Bodies

2009 ◽  
Vol 52 (3) ◽  
pp. 361-365 ◽  
Author(s):  
Fejes Tóth Gábor
Keyword(s):  
Convex Body ◽  
Euclidean Space ◽  
Convex Bodies ◽  
Dimensional Euclidean Space ◽  
Classical Theorem ◽  
Lattice Arrangement

AbstractA classical theorem of Rogers states that for any convex body K in n-dimensional Euclidean space there exists a covering of the space by translates of K with density not exceeding n log n + n log log n + 5n. Rogers’ theorem does not say anything about the structure of such a covering. We show that for sufficiently large values of n the same bound can be attained by a covering which is the union of O(log n) translates of a lattice arrangement of K.

A problem concerning three-dimensional convex bodies

1953 ◽  
Vol 49 (1) ◽  
pp. 44-53 ◽  
Author(s):  
R. A. Rankin
Keyword(s):  
Convex Body ◽  
Euclidean Space ◽  
Distance Function ◽  
Interior Point ◽  
Convex Bodies ◽  
Three Dimensional ◽  
Dimensional Euclidean Space ◽  
Bounded Convex

1. The problem considered in this paper arose during an investigation of what Chabauty has called the ‘anomaly’ of convex bodies.Throughout the paper denotes a closed bounded convex body in three-dimensional Euclidean space , which is symmetric in the origin O and which contains O as an interior point. Such a body determines uniquely a distance-function f(x, y, z) which is defined and finite for each point (x, y, z) of and possesses the following properties

scholarly journals Approximation Properties of Random Polytopes Associated with Poisson Hyperplane Processes

2014 ◽  
Vol 46 (4) ◽  
pp. 919-936
Author(s):  
Daniel Hug ◽  
Rolf Schneider
Keyword(s):  
Convex Body ◽  
Euclidean Space ◽  
Hausdorff Distance ◽  
Dimensional Euclidean Space ◽  
Approximation Properties ◽  
Random Polytopes ◽  
Directional Distribution ◽  
Zero Cell

We consider a stationary Poisson hyperplane process with given directional distribution and intensity in d-dimensional Euclidean space. Generalizing the zero cell of such a process, we fix a convex body K and consider the intersection of all closed halfspaces bounded by hyperplanes of the process and containing K. We study how well these random polytopes approximate K (measured by the Hausdorff distance) if the intensity increases, and how this approximation depends on the directional distribution in relation to properties of K.

The anomaly of convex bodies

1953 ◽  
Vol 49 (1) ◽  
pp. 54-58 ◽  
Author(s):  
R. A. Rankin
Keyword(s):  
Euclidean Space ◽  
Real Number ◽  
Convex Bodies ◽  
Dimensional Euclidean Space ◽  
Real Numbers

I write X for the point (x1, x2, …, xn) of n-dimensional Euclidean space Rn. The coordinates x1, x2, …, xn are real numbers. The origin (0, 0,…, 0) is denoted by O. If t is a real number, tX denotes the point (tx1, tx2, …, txn); in particular, − X is the point (−x1, −x2,…, −xn). Also X + Y denotes the point {x1 + y1, x2 + y2, …, xn + yn).

scholarly journals A characterisation of the ellipsoid

1970 ◽  
Vol 11 (4) ◽  
pp. 385-394 ◽  
Author(s):  
P. W. Aitchison
Keyword(s):  
Euclidean Space ◽  
Quadratic Forms ◽  
Convex Bodies ◽  
Positive Definite ◽  
Dimensional Euclidean Space ◽  
Major Result

The ellipsoid is characterised among all convex bodies in n-dimensional Euclidean space, Rn, by many different properties. In this paper we give a characterisation which generalizes a number of previous results mentioned in [2], p. 142. The major result will be used, in a paper yet to be published, to prove some results concerning generalizations of the Minkowski theory of reduction of positive definite quadratic forms.

scholarly journals Faces with given directions in anisotropic Poisson hyperplane mosaics

2011 ◽  
Vol 43 (2) ◽  
pp. 308-321 ◽  
Author(s):  
Daniel Hug ◽  
Rolf Schneider
Keyword(s):  
Euclidean Space ◽  
Crucial Role ◽  
Conditional Distribution ◽  
Convex Bodies ◽  
Dimensional Euclidean Space ◽  
Different Types ◽  
Typical Cell ◽  
Zero Cell ◽  
Asymptotic Problem

For stationary Poisson hyperplane tessellations in d-dimensional Euclidean space and a dimension k ∈ {1, …, d}, we investigate the typical k-face and the weighted typical k-face (weighted by k-dimensional volume), without isotropy assumptions on the tessellation. The case k = d concerns the previously studied typical cell and zero cell, respectively. For k < d, we first find the conditional distribution of the typical k-face or weighted typical k-face, given its direction. Then we investigate how the shapes of the faces are influenced by assumptions of different types: either via containment of convex bodies of given volume (including a new result for k = d), or, for weighted typical k-faces, in the spirit of D. G. Kendall's asymptotic problem, suitably generalized. In all these results on typical or weighted typical k-faces with given direction space L, the Blaschke body of the section process of the underlying hyperplane process with L plays a crucial role.

A separation theorem for totally-sewn 4-polytopes

2015 ◽  
Vol 52 (3) ◽  
pp. 386-422
Author(s):  
T. Bisztriczky ◽  
F. Fodor
Keyword(s):  
Convex Body ◽  
Euclidean Space ◽  
Interior Point ◽  
Separation Theorem ◽  
Dimensional Euclidean Space ◽  
Separation Problem ◽  
Minimum Number

The Separation Problem, originally posed by K. Bezdek in [1], asks for the minimum number s(O, K) of hyperplanes needed to strictly separate an interior point O in a convex body K from all faces of K. It is conjectured that s(O, K) ≦ 2d in d-dimensional Euclidean space. We prove this conjecture for the class of all totally-sewn neighbourly 4-dimensional polytopes.

On Measures of Symmetry of Convex Bodies

1965 ◽  
Vol 17 ◽  
pp. 497-504 ◽  
Author(s):  
G. D. Chakerian ◽  
S. K. Stein
Keyword(s):  
Convex Body ◽  
Convex Set ◽  
Compact Convex ◽  
Convex Bodies ◽  
Dimensional Euclidean Space ◽  
Maximum Volume ◽  
Compact Convex Set ◽  
Centrally Symmetric ◽  
Measures Of Symmetry ◽  
Interior Points

Let K be a convex body (compact, convex set with interior points) in n-dimensional Euclidean space En, and let V(K) denote the volume of K. Let K′ be a centrally symmetric body of maximum volume contained in K (in fact, K′ is unique; see 2 or 9), and definec(K) = V(K′)/V(K)Letc(n) = inf{c(K) : K ⊂ En}.

Inequalities for random flats meeting a convex body

1985 ◽  
Vol 22 (03) ◽  
pp. 710-716 ◽  
Author(s):  
Rolf Schneider
Keyword(s):  
Convex Body ◽  
Euclidean Space ◽  
Finite Number ◽  
Random Point ◽  
Dimensional Euclidean Space ◽  
Random Direction ◽  
Higher Dimensional

We choose a uniform random point in a given convex bodyKinn-dimensional Euclidean space and through that point the secant ofKwith random direction chosen independently and isotropically. Given the volume ofK, the expectation of the length of the resulting random secant ofKwas conjectured by Enns and Ehlers [5] to be maximal ifKis a ball. We prove this, and we also treat higher-dimensional sections defined in an analogous way. Next, we consider a finite number of independent isotropic uniform random flats meetingK, and we prove that certain geometric probabilities connected with these again become maximal whenKis a ball.

scholarly journals Two finiteness theorems in the Minkowski theory of reduction

1972 ◽  
Vol 14 (3) ◽  
pp. 336-351 ◽  
Author(s):  
P. W. Aitchison
Keyword(s):  
Convex Body ◽  
Euclidean Space ◽  
Quadratic Forms ◽  
Positive Definite ◽  
Dimensional Euclidean Space ◽  
Symmetric Convex Body ◽  
Symmetric Convex ◽  
Reduction Theory ◽  
Positive Definite Quadratic Form ◽  
Finiteness Theorems

Minkowski proved two important finiteness theorems concerning the reduction theory of positive definite quadratic forms (see [6], p. 285 or [7], §8 and §10). A positive definite quadratic form in n variables may be considered as an ellipsoid in n-dimensional Euclidean space, Rn, and then the two results can be investigated more generally by replacing the ellipsoid by any symmetric convex body in Rn. We show here that when n≧3 the two finiteness theorems hold only in the case of the ellipsoid. This is equivalent to showing that Minkowski's results do not hold in a general Minkowski space, namely in a euclidean space where the unit ball is a general symmetric convex body instead of the sphere or ellipsoid.

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