scholarly journals Two inequalities for convex sets in the plane

1978 ◽  
Vol 19 (1) ◽  
pp. 131-133 ◽  
Author(s):  
P.R. Scott
Keyword(s):  
Euclidean Plane ◽  
Convex Sets ◽  
Convex Set ◽  
Image Position ◽  
Closed Convex Set

Let K be a bounded, closed, convex set in the euclidean plane having diameter d, width w, inradius r, and circumradius R. We show thatandwhere both these inequalities are best possible.

scholarly journals A family of inequalities for convex sets

1979 ◽  
Vol 20 (2) ◽  
pp. 237-245 ◽  
Author(s):  
P.R. Scott
Keyword(s):  
Euclidean Plane ◽  
Convex Sets ◽  
Convex Set ◽  
Upper Bounds ◽  
Image Position ◽  
Closed Convex Set

Let K be a bounded, closed convex set in the euclidean plane. We denote the diameter, width, perimeter, area, inradius, and circumradius of K by d, w, p, A, r, and R respectively. We establish a number of best possible upper bounds for (w−2r)d, (w−2r)R,(w−2r)p, (w−2r)A in terms of w and r. Examples are:

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:

Points of egress in problems of Hamiltonian dynamics

1991 ◽  
Vol 109 (2) ◽  
pp. 405-417 ◽  
Author(s):  
C. J. Amick ◽  
J. F. Toland
Keyword(s):  
Convex Set ◽  
Hamiltonian Dynamics ◽  
Empty Interior ◽  
Natural Assumption ◽  
Initial Value ◽  
Lipschitz Continuous ◽  
Convexity Assumptions ◽  
Image Position ◽  
Closed Convex Set ◽  
Well Posed

First we consider an elementary though delicate question about the trajectory in ℝn of a particle in a conservative field of force whose dynamics are governed by the equationHere the potential function V is supposed to have Lipschitz continuous first derivative at every point of ℝn. This is a natural assumption which ensures that the initial-value problem is well-posed. We suppose also that there is a closed convex set C with non-empty interior C° such that V ≥ 0 in C and V = 0 on the boundary ∂C of C. It is noteworthy that we make no assumptions about the degeneracy (or otherwise) of V on ∂C (i.e. whether ∇V = 0 on ∂C, or not); thus ∂C is a Lipschitz boundary because of its convexity but not necessarily any smoother in general. We remark also that there are no convexity assumptions about V and nothing is known about the behaviour of V outside C.

SEPARATION OF CONVEX SETS IN EXTENDED NORMED SPACES

2015 ◽  
Vol 99 (2) ◽  
pp. 145-165 ◽  
Author(s):  
G. BEER ◽  
J. VANDERWERFF
Keyword(s):  
Convex Sets ◽  
Convex Set ◽  
Separation Theorems ◽  
Normed Spaces ◽  
Finite Dimensional ◽  
Weakly Closed ◽  
Separation Of Convex Sets ◽  
Real Linear ◽  
Closed Convex Set ◽  
Special Case

We give continuous separation theorems for convex sets in a real linear space equipped with a norm that can assume the value infinity. In such a space, it may be impossible to continuously strongly separate a point $p$ from a closed convex set not containing $p$, that is, closed convex sets need not be weakly closed. As a special case, separation in finite-dimensional extended normed spaces is considered at the outset.

Core-preserving transformations of a vector space

1953 ◽  
Vol 49 (1) ◽  
pp. 15-25
Author(s):  
F. F. Bonsall
Keyword(s):  
Vector Space ◽  
Convex Set ◽  
Complex Numbers ◽  
Abstract Theory ◽  
The Core ◽  
Bounded Complex ◽  
Complex Sequences ◽  
Image Position ◽  
Closed Convex Set ◽  
Bounded Sequences

In the classical theory (3) due to Knopp, Agnew and others, the core K(x) of a sequence x = {ξn} of complex numbers is defined by where En(x) is the smallest closed convex set containing all ξk with k ≥ n. A matrix transformation T is said to be a core-preserving transformation ifholds for all sequences x. T is core-preserving for bounded sequences if (1·1) holds for all bounded sequences x. It is readily proved that K(x) is the set of complex numbers ζ such thatfor all complex numbers α (). Now is a sub-additive, positive-homogeneous real-valued functional defined on the vector space of bounded complex sequences. This suggests the construction of an abstract theory on the following lines.

Cushions, cigars and diamonds: an area-perimeter problem for symmetric ovals

1979 ◽  
Vol 85 (1) ◽  
pp. 1-16 ◽  
Author(s):  
H. T. Croft
Keyword(s):  
Upper Bound ◽  
Natural Condition ◽  
Lattice Point ◽  
Convex Set ◽  
General Setting ◽  
Lattice Points ◽  
Two Dimensional ◽  
Algebraic Expressions ◽  
Image Position ◽  
Closed Convex Set

P. R. Scott (1) has asked which two-dimensional closed convex set E, centro-symmetric in the origin O, and containing no other Cartesian lattice-point in its interior, maximizes the ratio A/P, where A, P are the area, perimeter of E; he conjectured that the answer is the ‘rounded square’ (‘cushion’ in what follows), described below. We shall prove this, indeed in a more general setting, by seeking to maximizewhere κ is a parameter (0 < κ < 2); the set of admissible E is those E centro-symmetric in 0 that do not contain in their interior certain fixed lattice-points. There are two problems, the unrestricted one , where there is no given upper bound on A (it will become apparent that this problem only has a finite answer when κ ≥ 1) and the restricted one , when one is given a bound B and we must have A ≤ B. Special interest attaches to the case B = 4, both because of Minkowski's theorem: any E symmetric in O and containing no other lattice-point has area at most 4; and because it turns out that it is a ‘natural’ condition: the algebraic expressions simplify to a remarkable extent. Hence in what follows, the ‘restricted case ’ shall mean A ≤ 4.

scholarly journals The weak drop property on closed convex sets

1994 ◽  
Vol 56 (1) ◽  
pp. 125-130
Author(s):  
Pei-Kee Lin ◽  
Xintai Yu
Keyword(s):  
Convex Sets ◽  
Convex Set ◽  
Nonempty Interior ◽  
Weakly Compact ◽  
Closed Set ◽  
Reflexive Space ◽  
Closed Convex Set

AbstractRecall a closed convex set C is said to have the weak drop property if for every weakly sequentially closed set A disjoint from C there exists x ∈ A such that co({x} ∩ C) ∪ A = {x}. Giles and Kutzarova proved that every bounded closed convex set with the weak drop property is weakly compact. In this article, we show that if C is an unbounded closed convex set of X with the weak drop property, then C has nonempty interior and X is a reflexive space.

scholarly journals Minimal requirements for Minkowski's theorem in the plane II

1980 ◽  
Vol 22 (2) ◽  
pp. 275-283
Author(s):  
J.R. Arkinstall
Keyword(s):  
Euclidean Plane ◽  
Convex Set ◽  
Integer Lattice ◽  
Equal Area ◽  
Closed Convex Set

Let K be a closed convex set in the Euclidean plane, with area A(K), which contains in its interior only one point 0 of the integer lattice. If K has other than one or three chords through 0 of one of the following types, it is shown that A(K) ≤ 4, while if K has three of one type, A(K) ≤ 4.5. The types of chords considered are chords which partition K into two regions of equal area, chords which lie midway between parallel supporting lines of K, and chords such that K is invariant under reflection in them. The results are generalised to any lattice in the plane.

scholarly journals External analysis of boundary points of convex sets: illumination and visibility

2009 ◽  
Vol 105 (2) ◽  
pp. 265 ◽  
Author(s):  
José Pedro Moreno ◽  
Alberto Seeger
Keyword(s):  
Normed Space ◽  
Convex Sets ◽  
Convex Set ◽  
Boundary Points ◽  
External Analysis ◽  
Closed Convex Set

The purpose of this work is studying the geometry of the boundary $\partial K$ of a solid closed convex set $K$ in a normed space. In a recent paper of ours, such a study has been carried out with the help of supporting cones and drops. Now, illuminated sets and visible sets are the main tools of analysis.

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.

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