scholarly journals Support and separation properties of convex sets in finite dimension

2021 ◽  
Vol 36 (2) ◽  
pp. 241-278
Author(s):  
Valeriu Soltan
Keyword(s):  
Euclidean Space ◽  
Convex Sets ◽  
Finite Dimension ◽  
Convex Bodies ◽  
Detailed Account ◽  
Dimensional Euclidean Space ◽  
Arbitrary Convex

This is a survey on support and separation properties of convex sets in the n-dimensional Euclidean space. It contains a detailed account of existing results, given either chronologically or in related groups, and exhibits them in a uniform way, including terminology and notation. We first discuss classical Minkowski’s theorems on support and separation of convex bodies, and next describe various generalizations of these results to the case of arbitrary convex sets, which concern bounding and asymptotic hyperplanes, and various types of separation by hyperplanes, slabs, and complementary convex sets.

scholarly journals A Helly-type theorem on a sphere

1967 ◽  
Vol 7 (3) ◽  
pp. 323-326 ◽  
Author(s):  
M. J. C. Baker
Keyword(s):  
Euclidean Space ◽  
Convex Sets ◽  
Type Theorem ◽  
Dimensional Euclidean Space ◽  
Dimensional Sphere ◽  
Strongly Convex ◽  
Set Of Points

The purpose of this paper is to prove that if n+3, or more, strongly convex sets on an n dimensional sphere are such that each intersection of n+2 of them is empty, then the intersection of some n+1 of them is empty. (The n dimensional sphere is understood to be the set of points in n+1 dimensional Euclidean space satisfying x21+x22+ …+x2n+1 = 1.)

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.

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.

scholarly journals Cone asymptotes of convex sets

2021 ◽  
Vol 36 (1) ◽  
pp. 81-98
Author(s):  
V. Soltan
Keyword(s):  
Euclidean Space ◽  
Convex Sets ◽  
Dimensional Euclidean Space

Based on the notion of plane asymptote, we introduce the new concept of cone asymptote of a set in the n-dimensional Euclidean space. We discuss the existence and describe some families of cone asymptotes.

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.

scholarly journals On the Irreducibility of Convex Bodies

1959 ◽  
Vol 11 ◽  
pp. 256-261 ◽  
Author(s):  
A. C. Woods
Keyword(s):  
Euclidean Space ◽  
Convex Bodies ◽  
Dimensional Euclidean Space ◽  
Star Body ◽  
Closed Set ◽  
Absolute Value ◽  
Linearly Independent ◽  
The Absolute ◽  
Set Of Points ◽  
Rational Integral

We select a Cartesian co-ordinate system in ndimensional Euclidean space Rn with origin 0 and employ the usual pointvector notation.By a lattice Λ in Rn we mean the set of all rational integral combinations of n linearly independent points X1, X2, … , Xn of Rn. The points X1 X2, … , Xn are said to form a basis of Λ. Let {X1, X2, … , Xn) denote the determinant formed when the co-ordinates of Xi are taken in order as the ith row of the determinant for i = 1,2, … , n. The absolute value of this determinant is called the determinant d(Λ) of Λ. It is well known that d(Λ) is independent of the particular basis one takes for Λ.A star body in Rn is a closed set of points K such that if X ∈ K then every point of the form tX where — 1 < t < 1 is an inner point of K.

Some extremal properties of convex sets

1975 ◽  
Vol 77 (3) ◽  
pp. 515-524 ◽  
Author(s):  
J. N. Lillington
Keyword(s):  
Euclidean Space ◽  
Lower Bounds ◽  
Convex Sets ◽  
Compact Convex ◽  
Upper And Lower Bounds ◽  
Arbitrary Convex ◽  
Extremal Properties ◽  
Convex Subsets

In this paper we shall suppose that all convex sets are compact convex subsets of Euclidean space En. We shall be concerned in producing upper and lower bounds for the ‘total edge lengths’ of simplices which are contained in or contain arbitrary convex sets in terms of the inradii and circumradii of these sets. However, before proceeding further, we shall introduce some notation and give some motivation for this work.

scholarly journals Faces with given directions in anisotropic Poisson hyperplane mosaics

2011 ◽  
Vol 43 (02) ◽  
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 &lt; 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.

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