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).

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 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 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 Decision problems for linear recurrences involving arbitrary real numbers

2021 ◽  
Vol Volume 17, Issue 3 ◽  
Author(s):  
Eike Neumann
Keyword(s):  
Euclidean Space ◽  
Real Number ◽  
Lebesgue Measure ◽  
Decision Problems ◽  
Real Numbers ◽  
Linear Recurrences ◽  
Initial Values ◽  
Problem Instances ◽  
Low Order

We study the decidability of the Skolem Problem, the Positivity Problem, and the Ultimate Positivity Problem for linear recurrences with real number initial values and real number coefficients in the bit-model of real computation. We show that for each problem there exists a correct partial algorithm which halts for all problem instances for which the answer is locally constant, thus establishing that all three problems are as close to decidable as one can expect them to be in this setting. We further show that the algorithms for the Positivity Problem and the Ultimate Positivity Problem halt on almost every instance with respect to the usual Lebesgue measure on Euclidean space. In comparison, the analogous problems for exact rational or real algebraic coefficients are known to be decidable only for linear recurrences of fairly low order.

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.

scholarly journals On lattice points in n -dimensional star bodies I. Existence theorems

1946 ◽  
Vol 187 (1009) ◽  
pp. 151-187 ◽  
Keyword(s):  
Euclidean Space ◽  
Existence Theorems ◽  
Lattice Points ◽  
Infinite Group ◽  
Dimensional Euclidean Space ◽  
Star Body ◽  
Linear Transformations ◽  
Real Numbers ◽  
Negative Function ◽  
Star Bodies

Let F ( X ) = F ( x 1 ,..., x n ) be a continuous non-negative function of X satisfying F ( tX ) = | t | F ( X ) for all real numbers t . The set K in n -dimensional Euclidean space R n defined by F ( X )⩽ 1 is called a star body. The author studies the lattices Λ in R n which are of minimum determinant and have no point except (0, ..., 0) inside K . He investigates how many points of such lattices lie on, or near to, the boundary of K , and considers in detail the case when K admits an infinite group of linear transformations into itself.

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.

Two problems on convex bodies

1962 ◽  
Vol 58 (1) ◽  
pp. 1-7 ◽  
Author(s):  
H. T. Croft
Keyword(s):  
New York ◽  
Euclidean Space ◽  
Convex Surface ◽  
Convex Bodies ◽  
Three Dimensional ◽  
Plane Section ◽  
Dimensional Euclidean Space ◽  
Centrally Symmetric ◽  
Mathematical Problems ◽  
Symmetric Set

We solve two problems on convex bodies stated on p. 38 of S. M. Ulam's book, A collection of mathematical problems (New York, 1960).Problem 1. This problem is due to Mazur. In three-dimensional Euclidean space there is given a convex surface W and a point O in its interior. Consider the set V of all points P defined by the requirement that the length of the interval OP is equal to the area of the plane section of W through O and perpendicular to OP. Is the (centrally symmetric) set V a convex surface?

Lattice Octahedra

1960 ◽  
Vol 12 ◽  
pp. 297-302 ◽  
Author(s):  
L. J. Mordell
Keyword(s):  
Euclidean Space ◽  
Dimensional Euclidean Space ◽  
Real Numbers ◽  
Linearly Independent ◽  
Image Position

Let Ai, A2, … , An be n linearly independent points in n-dimensional Euclidean space of a lattice Λ. The points ± A1, ±A2, . . , ±An define a closed n-dimensional octahedron (or “cross poly tope“) K with centre at the origin O. Our problem is to find a basis for the lattices Λ which have no points in K except ±A1, ±A2, … , ±An.Let the position of a point P in space be defined vectorially by1where the p are real numbers. We have the following results.When n = 2, it is well known that a basis is2When n = 3, Minkowski (1) proved that there are two types of lattices, with respective bases3When n = 4, there are six essentially different bases typified by A1, A2, A3 and one of4In all expressions of this kind, the signs are independent of each other and of any other signs. This result is a restatement of a result by Brunngraber (2) and a proof is given by Wolff (3).

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