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.

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

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.

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.

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.

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.

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

2014 ◽  
Vol 46 (04) ◽  
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 ind-dimensional Euclidean space. Generalizing the zero cell of such a process, we fix a convex bodyKand consider the intersection of all closed halfspaces bounded by hyperplanes of the process and containingK. We study how well these random polytopes approximateK(measured by the Hausdorff distance) if the intensity increases, and how this approximation depends on the directional distribution in relation to properties ofK.

Inequalities for random flats meeting a convex body

1985 ◽  
Vol 22 (3) ◽  
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 body K in n-dimensional Euclidean space and through that point the secant of K with random direction chosen independently and isotropically. Given the volume of K, the expectation of the length of the resulting random secant of K was conjectured by Enns and Ehlers [5] to be maximal if K is 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 meeting K, and we prove that certain geometric probabilities connected with these again become maximal when K is a ball.

On Vitali's Theorem for Groups of Motions of Euclidean Space

1999 ◽  
Vol 6 (4) ◽  
pp. 323-334
Author(s):  
A. Kharazishvili
Keyword(s):  
Euclidean Space ◽  
Dimensional Euclidean Space ◽  
Finite Dimensional

Abstract We give a characterization of all those groups of isometric transformations of a finite-dimensional Euclidean space, for which an analogue of the classical Vitali theorem [Sul problema della misura dei gruppi di punti di una retta, 1905] holds true. This characterization is formulated in purely geometrical terms.

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