The Steiner Point of a Convex Polytope

1966 ◽  
Vol 18 ◽  
pp. 1294-1300 ◽  
Author(s):  
G. C. Shephard
Keyword(s):  
Euclidean Space ◽  
Extremal Problem ◽  
Convex Sets ◽  
Convex Set ◽  
Convex Polytope ◽  
Steiner Point ◽  
Dimensional Euclidean Space ◽  
Vector Addition ◽  
Image Position ◽  
Bounded Convex

Associated with each bounded convex set K in n-dimensional euclidean space En is a point s(K) known as its Steiner point. First considered by Steiner in 1840 (6, p. 99) in connection with an extremal problem for convex regions, this point has been found useful in some recent investigations (for example, 4) because of the linearity property1Addition on the left is vector addition of convex sets.

scholarly journals Lattice points in a convex Set of given width

1976 ◽  
Vol 21 (4) ◽  
pp. 504-507 ◽  
Author(s):  
G. B. Elkington ◽  
J. Hammer
Keyword(s):  
Euclidean Space ◽  
Minimum Distance ◽  
Convex Set ◽  
Lattice Points ◽  
Dimensional Euclidean Space ◽  
Integral Lattice ◽  
Supporting Hyperplanes ◽  
Bounded Convex

Let S be a closed bounded convex set in d-dimensional Euclidean space Ed. The width w(S) of S is the minimum distance between supporting hyperplanes of S, and L(S) is the number of integral lattice points in the interior of S.

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.)

Properties of Harmonic Functions of Three Real Variables Given by Bergman-Whittaker Operators

1963 ◽  
Vol 15 ◽  
pp. 157-168 ◽  
Author(s):  
Josephine Mitchell
Keyword(s):  
Euclidean Space ◽  
Harmonic Functions ◽  
Three Dimensional ◽  
Dimensional Euclidean Space ◽  
Rectifiable Curve ◽  
Image Position

Let be a closed rectifiable curve, not going through the origin, which bounds a domain Ω in the complex ζ-plane. Let X = (x, y, z) be a point in three-dimensional euclidean space E3 and setThe Bergman-Whittaker operator defined by

Additive functions of intervals and Hausdorff measure

1946 ◽  
Vol 42 (1) ◽  
pp. 15-23 ◽  
Author(s):  
P. A. P. Moran
Keyword(s):  
Lower Bound ◽  
Euclidean Space ◽  
Hausdorff Measure ◽  
Convex Sets ◽  
Infinite Sequence ◽  
Small Positive Number ◽  
Additive Functions ◽  
Image Position ◽  
Bounded Sets ◽  
Increasing Function

Consider bounded sets of points in a Euclidean space Rq of q dimensions. Let h(t) be a continuous increasing function, positive for t>0, and such that h(0) = 0. Then the Hausdroff measure h–mE of a set E in Rq, relative to the function h(t), is defined as follows. Let ε be a small positive number and suppose E is covered by a finite or enumerably infinite sequence of convex sets {Ui} (open or closed) of diameters di less than or equal to ε. Write h–mεE = greatest lower bound for any such sequence {Ui}. Then h–mεE is non-decreasing as ε tends to zero. We define

Vector sums of Valentine convex sets

1982 ◽  
Vol 92 (1) ◽  
pp. 17-19
Author(s):  
H. G. Eggleston
Keyword(s):  
Euclidean Space ◽  
Convex Sets ◽  
Convex Set ◽  
Dimensional Space ◽  
End Points ◽  
3 Dimensional ◽  
Vector Sum ◽  
Open Question

SummaryA set X in Euclidean space is Valentine n–convex, or simply n–convex' if it has the following property. If X contains a subset Y consisting of n distinct points then X also contains the points of at least one segment with end points in Y. We show here that the vector sum of two plane compact 3-convex sets is 5-convex (which complements the result of I. D. Calvert(1) that the intersection of two plane compact 3-convex sets is 5-convex) and that the vector sum of a plane connected compact 3-convex set with itself is 4-convex. These results are not true in 4 dimensional space. It is an open question whether or not they are true in 3-dimensional space.

Valentine convexity in n dimensions

1976 ◽  
Vol 80 (2) ◽  
pp. 223-228
Author(s):  
H. G. Eggleston
Keyword(s):  
Euclidean Space ◽  
Convex Sets ◽  
Convex Set ◽  
Two Dimensions ◽  
Lower Dimension

A subset X of Euclidean space such that if a, b, c are points of X then at least one of the segments joining two of them lies in X, is said to be V-convex. Valentine (4) showed that in two dimensions a compact V-convex set is the union of at most three convex sets. We show here that if the set of star centres of X is of lower dimension than X and X is a compact V-convex set then it is the union of at most two convex sets.

scholarly journals Further inequalities for convex sets with lattice point constraints in the plane

1980 ◽  
Vol 21 (1) ◽  
pp. 7-12 ◽  
Author(s):  
P.R. Scott
Keyword(s):  
Lattice Point ◽  
Convex Sets ◽  
Convex Set ◽  
Integral Lattice ◽  
Image Position ◽  
Closed Convex Set

Let K be a bounded closed convex set in the plane containing no points of the integral lattice in its interior and having width w, area A, perimeter p and circumradius R. The following best possible inequalities are established:

Multiple points for symmetric Lévy processes

1978 ◽  
Vol 83 (1) ◽  
pp. 83-90 ◽  
Author(s):  
John Hawkes
Keyword(s):  
Markov Process ◽  
Euclidean Space ◽  
Lévy Process ◽  
Lévy Processes ◽  
Transition Function ◽  
Levy Processes ◽  
Levy Process ◽  
Dimensional Euclidean Space ◽  
Multiple Points ◽  
Image Position

Let Xt be a Lévy process in Rd, d-dimensional euclidean space. That is X is a Markov process whose transition function satisfies

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.

A Decomposition for Sets Having a Segment Convexity Property

1980 ◽  
Vol 32 (1) ◽  
pp. 21-26
Author(s):  
Marilyn Breen
Keyword(s):  
Euclidean Space ◽  
Convex Sets ◽  
Convex Set ◽  
Finite Union ◽  
Combinatorial Property ◽  
Convexity Property ◽  
Convexity Condition ◽  
Line Segments ◽  
Isolated Points ◽  
Decomposition Theorems

Let 5 be a subset of Euclidean space. The set 5 is said to be m-convex, m ≥ 2, if and only if for every m distinct points of S, at least one of the line segments determined by these points lies in 5. Clearly any union of m mdash — 1 convex sets will be m-convex, yet the converse is false. However, several decomposition theorems have been proved which allow us to write any closed planar m-convex set as a finite union of convex sets, and actual bounds for the decomposition in terms of m have been obtained ([6], [4], [3]). Moreover, with the restriction that (int cl S) ∼ S contain no isolated points, an arbitrary planar m-convex set S may be decomposed into a finite union of convex sets ([1].Here we strengthen the m-convexity condition to define an analogous combinatorial property for segments.

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