The Continuity of the Visibility Function on a Starshaped Set

1972 ◽  
Vol 24 (5) ◽  
pp. 989-992 ◽  
Author(s):  
Gerald Beer
Keyword(s):  
Euclidean Space ◽  
Sufficient Conditions ◽  
Outer Measure ◽  
Visibility Function ◽  
Starshaped Set ◽  
Measurable Set

The visibility function assigns to each point x of a fixed measurable set E in a Euclidean space En the Lebesgue outer measure of S(x), the set {y : rx + (1 — r)y ∊ E for every r in [0, 1]}.The purpose of this paper is to determine sufficient conditions for the continuity of the function on the interor of a starshaped set.

scholarly journals Horosphere slab separation theorems in manifolds without conjugate points

2019 ◽  
Vol 27 (1) ◽  
Author(s):  
Sameh Shenawy
Keyword(s):  
Euclidean Space ◽  
Hyperbolic Space ◽  
Existence And Uniqueness ◽  
Convex Set ◽  
Sufficient Conditions ◽  
Conjugate Points ◽  
Separation Theorems ◽  
Farthest Points ◽  
Simply Connected ◽  
Convex Subsets

Abstract Let $\mathcal {W}^{n}$ W n be the set of smooth complete simply connected n-dimensional manifolds without conjugate points. The Euclidean space and the hyperbolic space are examples of these manifolds. Let $W\in \mathcal {W}^{n}$ W ∈ W n and let A and B be two convex subsets of W. This note aims to investigate separation and slab horosphere separation of A and B. For example,sufficient conditions on A and B to be separated by a slab of horospheres are obtained. Existence and uniqueness of foot points and farthest points of a convex set A in $W\in \mathcal {W}$ W ∈ W are considered.

A measureless one-dimensional set

1954 ◽  
Vol 50 (3) ◽  
pp. 391-393
Author(s):  
H. G. Eggleston
Keyword(s):  
Hausdorff Dimension ◽  
Euclidean Space ◽  
Single Point ◽  
Outer Measure ◽  
One Dimensional ◽  
Infinite Measure

It has been known for some time that there are sets in Euclidean space which are of infinite measure in a certain Hausdorff dimension, say α, and yet which contain no subsets that are of finite positive a measure. The properties of such a set, say X, could be developed in much the same way as those of sets which are of positive finite α measure, if there existed a Caratheodory outer measure Γ, defined over the subsets of X, which was such that ∞ > Γ(X) > 0, and for any subset Y of X, Λα (Y) = 0 implied Γ (Y) = O. The object of this note is to show that there are sets for which no such outer measure exists. It is shown that a set defined by Sierpinski (7) is one-dimensional and is such that any Caratheodory outer measure defined over its subsets either takes infinite values or is identically zero or is not zero for all subsets that consist of a single point. It has been remarked by Prof. Besicovitch that these properties imply that the set is measurable with respect to any Caratheodory outer measure defined over subsets of the plane, and in fact any subset is measurable with respect to any such outer measure.

Semi-E-Preinvex Functions

2012 ◽  
pp. 677-683
Author(s):  
Yu-Ru Syau ◽  
E. Stanley Lee
Keyword(s):  
Nonlinear Programming ◽  
Euclidean Space ◽  
Convex Functions ◽  
Sufficient Conditions ◽  
Dimensional Euclidean Space ◽  
Quasiconvex Functions ◽  
Invex Set ◽  
Preinvex Functions ◽  
The Relationship ◽  
Class Of Functions

A class of functions called semi-E-preinvex functions is defined as a generalization of semi-E-convex functions. Similarly, the concept of semi-E-quasiconvex functions is also generalized to semi-E-prequasiinvex functions. Properties of these proposed classes are studied, and sufficient conditions for a nonempty subset of the n-dimensional Euclidean space to be an E-convex or E-invex set are given. The relationship between semi-E-preinvex and E-preinvex functions are discussed along with results for the corresponding nonlinear programming problems.

Semi-E-Preinvex Functions

2010 ◽  
Vol 1 (3) ◽  
pp. 31-39
Author(s):  
Yu-Ru Syau ◽  
E. Stanley Lee
Keyword(s):  
Nonlinear Programming ◽  
Euclidean Space ◽  
Nonempty Subset ◽  
Sufficient Conditions ◽  
Dimensional Euclidean Space ◽  
Quasiconvex Functions ◽  
Invex Set ◽  
Preinvex Functions ◽  
The Relationship ◽  
Class Of Functions

A class of functions called semi--preinvex functions is defined as a generalization of semi--convex functions. Similarly, the concept of semi--quasiconvex functions is also generalized to semi--prequasiinvex functions. Properties of these proposed classes are studied, and sufficient conditions for a nonempty subset of the -dimensional Euclidean space to be an -convex or -invex set are given. The relationship between semi--preinvex and -preinvex functions are discussed along with results for the corresponding nonlinear programming problems.

Mean size-and-shapes and mean shapes: a geometric point of view

1995 ◽  
Vol 27 (1) ◽  
pp. 44-55 ◽  
Author(s):  
Hulling Le
Keyword(s):  
Euclidean Space ◽  
Sufficient Conditions ◽  
Point Of View ◽  
Shape Spaces ◽  
Size And Shape ◽  
Geometric Point ◽  
Theoretic Point ◽  
Mean Size

Unlike the means of distributions on a euclidean space, it is not entirely clear how one should define the means of distributions on the size-and-shape or shape spaces of k labelled points in ℝm since these spaces are all curved. In this paper, we discuss, from a shape-theoretic point of view, some questions which arise in practice while using procrustean methods to define mean size-and-shapes or shapes. We obtain sufficient conditions for such means to be unique and for the corresponding generalized procrustean algorithms to converge to them. These conditions involve the curvature of the size-and-shape or shape spaces and are much less restrictive than asking for the data to be concentrated.

scholarly journals A New Approach to Mannheim Curve in Euclidean 3-Space

2021 ◽  
Vol 54 ◽  
Author(s):  
Ali UÇUM ◽  
Çetin Camcı ◽  
Kazım İlarslan
Keyword(s):  
Euclidean Space ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Dimensional Euclidean Space ◽  
New Approach ◽  
3 Dimensional ◽  
Mannheim Curve ◽  
Necessary And Sufficient

In this article, a new approach is given for Mannheim curves in 3-dimensional Euclidean space. Thanks to this approach, the necessary and sufficient conditions including the known results have been obtained for a curve to be Mannheim curve in E³. In addition, related examples and graphs are given by showing that there can be Mannheim curves in Salkowski or anti-Salkowski curves as well as giving Mannheim mate curves, which are not in literature. Finally, the Mannheim partner curves are characterized in E³.

scholarly journals Reconstruction of the One-Dimensional Lebesgue Measure

2020 ◽  
Vol 28 (1) ◽  
pp. 93-104
Author(s):  
Noboru Endou
Keyword(s):  
Lebesgue Measure ◽  
Real Space ◽  
Outer Measure ◽  
Finite Additivity ◽  
One Dimensional ◽  
The Real ◽  
Measurable Set ◽  
The One ◽  
System 1 ◽  
Dimensional Lebesgue Measure

SummaryIn the Mizar system ([1], [2]), Józef Białas has already given the one-dimensional Lebesgue measure [4]. However, the measure introduced by Białas limited the outer measure to a field with finite additivity. So, although it satisfies the nature of the measure, it cannot specify the length of measurable sets and also it cannot determine what kind of set is a measurable set. From the above, the authors first determined the length of the interval by the outer measure. Specifically, we used the compactness of the real space. Next, we constructed the pre-measure by limiting the outer measure to a semialgebra of intervals. Furthermore, by repeating the extension of the previous measure, we reconstructed the one-dimensional Lebesgue measure [7], [3].

scholarly journals LEBESGUE’S CRITERION FOR INTEGRABILITY

EDUPEDIA ◽  
2018 ◽  
Vol 2 (1) ◽  
pp. 12
Author(s):  
Bayu Riyadhus Saputro ◽  
Sumaji .
Keyword(s):  
Lebesgue Measure ◽  
Measurable Function ◽  
Integrable Function ◽  
Measure Zero ◽  
Bounded Function ◽  
Simple Function ◽  
Outer Measure ◽  
Lebesgue Integral ◽  
Lebesgue Integrable Function ◽  
Measurable Set

Lebesgue integral was defined constructively by using outer measure, measurable set, measurable function and Lebesgue measure concept. Then it was constructed by using simple function. The requirement of a Lebesgue integrable function to be Riemann integrable is bounded function and the set of discontinuities of that function has measure zero.

scholarly journals Semi-symmetry of δ(2,2) Chen ideal submanifolds

Filomat ◽  
2015 ◽  
Vol 29 (3) ◽  
pp. 393-400
Author(s):  
Anica Pantic ◽  
Miroslava Petrovic-Torgaseva
Keyword(s):  
Euclidean Space ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Necessary And Sufficient

In this paper we discuss ?(2,2) Chen ideal submanifolds M4 in the Euclidean space E6, and we find the necessary and sufficient conditions under which such a submanifold M4 is semi-symmetric, i.e. it satisfies the condition R(X,Y)? R = 0.

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