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.

Independence for Sets of Topological Spheres

1991 ◽  
Vol 34 (4) ◽  
pp. 520-524
Author(s):  
Lewis Pakula ◽  
Sol Schwartzman
Keyword(s):  
Euclidean Space ◽  
Convex Sets ◽  
Compact Convex ◽  
Necessary Condition ◽  
Number Of Components

AbstractConsider a collection of topological spheres in Euclidean space whose intersections are essentially topological spheres. We find a bound for the number of components of the complement of their union and discuss conditions for the bound to be achieved. This is used to give a necessary condition for independence of these sets. A related conjecture of Griinbaum on compact convex sets is discussed.

On Certain Intersection Properties of Convex Sets

1951 ◽  
Vol 3 ◽  
pp. 272-275 ◽  
Author(s):  
V. L. Klee
Keyword(s):  
Euclidean Space ◽  
Interior Point ◽  
Convex Sets ◽  
Common Interior Point ◽  
Convex Subsets

A collection of n + 1 convex subsets of a Euclidean space E will be called an n-set in E provided each n of the sets have a common interior point although the intersection of all n + 1 interiors is empty. It is well-known that if {C0,C1} is a 1-set, then C0 and C1 can be separated by a hyperplane.

COVERING A SET OF POINTS WITH A MINIMUM NUMBER OF TURNS

2004 ◽  
Vol 14 (01n02) ◽  
pp. 105-114 ◽  
Author(s):  
MICHAEL J. COLLINS
Keyword(s):  
Euclidean Space ◽  
Lower Bounds ◽  
Piecewise Linear ◽  
Upper And Lower Bounds ◽  
Linear Path ◽  
Change Direction ◽  
Finite Set ◽  
Minimum Number ◽  
Pass Through ◽  
Set Of Points

Given a finite set of points in Euclidean space, we can ask what is the minimum number of times a piecewise-linear path must change direction in order to pass through all of them. We prove some new upper and lower bounds for the rectilinear version of this problem in which all motion is orthogonal to the coordinate axes. We also consider the more general case of arbitrary directions.

scholarly journals On two functionals involving the maximum of the torsion function

2018 ◽  
Vol 24 (4) ◽  
pp. 1585-1604 ◽  
Author(s):  
Antoine Henrot ◽  
Ilaria Lucardesi ◽  
Gérard Philippin
Keyword(s):  
Lebesgue Measure ◽  
Lower Bounds ◽  
Convex Sets ◽  
Upper And Lower Bounds ◽  
Dirichlet Eigenvalue ◽  
Torsion Function ◽  
Open Set ◽  
First Dirichlet Eigenvalue ◽  
Finite Lebesgue Measure

In this paper we investigate upper and lower bounds of two shape functionals involving the maximum of the torsion function. More precisely, we consider T(Ω)∕(M(Ω)|Ω|) and M(Ω)λ1(Ω), where Ω is a bounded open set of ℝd with finite Lebesgue measure |Ω|, M(Ω) denotes the maximum of the torsion function, T(Ω) the torsion, and λ1(Ω) the first Dirichlet eigenvalue. Particular attention is devoted to the subclass of convex sets.

Killed Brownian motion and the Brunn–Minkowski inequalities

2012 ◽  
Vol 153 (1) ◽  
pp. 111-121
Author(s):  
HUILING LE
Keyword(s):  
Brownian Motion ◽  
Euclidean Space ◽  
Poisson Equation ◽  
Harmonic Functions ◽  
Convex Sets ◽  
Compact Convex ◽  
Compact Sets ◽  
Brownian Motions ◽  
The Poisson Equation ◽  
Killed Brownian Motions

AbstractWe construct triplets of killed Brownian motions to obtain the Brunn–Minkowski inequalities concerning the solutions of the equation (1/2)Δψ − h ψ = g on three interrelated compact sets in Euclidean space. These, in particular, include inequalities relating to the solutions of the Schrödinger equation and the Poisson equation on the three compact convex sets and an inequality relating to harmonic functions.

A Banach Space Whose Elements are Classes of Sets of Constant Width

1975 ◽  
Vol 18 (5) ◽  
pp. 679-689 ◽  
Author(s):  
J. E. Lewis
Keyword(s):  
Banach Space ◽  
Euclidean Space ◽  
Compact Subset ◽  
Compact Convex ◽  
Constant Width ◽  
The Real ◽  
Real Euclidean Space ◽  
Convex Subsets

Let K be a compact subset of the real Euclidean space En. We say that K has constant width if the distance between each pair of distinct parallel hyperplanes which support K is constant. The collection of all compact convex subsets of En which have constant width is denoted .

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 On the Generalized Distance Energy of Graphs

Mathematics ◽  
2019 ◽  
Vol 8 (1) ◽  
pp. 17 ◽  
Author(s):  
Abdollah Alhevaz ◽  
Maryam Baghipur ◽  
Hilal A. Ganie ◽  
Yilun Shang
Keyword(s):  
Lower Bounds ◽  
Complete Graph ◽  
Connected Graph ◽  
Distance Matrix ◽  
Wiener Index ◽  
Diagonal Matrix ◽  
Upper And Lower Bounds ◽  
Star Graph ◽  
Extremal Graphs ◽  
Generalized Distance

The generalized distance matrix D α ( G ) of a connected graph G is defined as D α ( G ) = α T r ( G ) + ( 1 − α ) D ( G ) , where 0 ≤ α ≤ 1 , D ( G ) is the distance matrix and T r ( G ) is the diagonal matrix of the node transmissions. In this paper, we extend the concept of energy to the generalized distance matrix and define the generalized distance energy E D α ( G ) . Some new upper and lower bounds for the generalized distance energy E D α ( G ) of G are established based on parameters including the Wiener index W ( G ) and the transmission degrees. Extremal graphs attaining these bounds are identified. It is found that the complete graph has the minimum generalized distance energy among all connected graphs, while the minimum is attained by the star graph among trees of order n.

scholarly journals Hadamard k-fractional inequalities of Fejér type for GA-s-convex mappings and applications

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Hui Lei ◽  
Gou Hu ◽  
Zhi-Jie Cao ◽  
Ting-Song Du
Keyword(s):  
Lower Bounds ◽  
Hypergeometric Functions ◽  
Upper And Lower Bounds ◽  
Weighted Inequalities ◽  
Convex Mappings ◽  
Special Means

Abstract The main aim of this paper is to establish some Fejér-type inequalities involving hypergeometric functions in terms of GA-s-convexity. For this purpose, we construct a Hadamard k-fractional identity related to geometrically symmetric mappings. Moreover, we give the upper and lower bounds for the weighted inequalities via products of two different mappings. Some applications of the presented results to special means are also provided.

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