scholarly journals Upper semicontinuity of the lamination hull

2018 ◽  
Vol 24 (4) ◽  
pp. 1503-1510
Author(s):  
Terence L.J. Harris
Keyword(s):  
Convex Hull ◽  
Upper Semicontinuity ◽  
Symmetric Matrices ◽  
Compact Set ◽  
Upper Semicontinuous ◽  
Rank One ◽  
Point Set

Let  K ⊆ ℝ2×2 be a compact set, let Krc be its rank-one convex hull, and let L (K) be its lamination convex hull. It is shown that the mapping K ↦ L̅(K̅) is not upper semicontinuous on the diagonal matrices in ℝ2×2, which was a problem left by Kolář. This is followed by an example of a 5-point set of 2 × 2 symmetric matrices with non-compact lamination hull. Finally, another 5-point set K is constructed, which has L (K) connected, compact and strictly smaller than Krc.

A NEW INVARIANT INFINITESIMAL METRIC

1990 ◽  
Vol 01 (01) ◽  
pp. 83-90 ◽  
Author(s):  
SHOSHICHI KOBAYASHI
Keyword(s):  
Convex Hull ◽  
Tangent Space ◽  
Complex Space ◽  
Upper Semicontinuity ◽  
Length Function ◽  
Nonzero Vector ◽  
Convexity Condition ◽  
Upper Semicontinuous

Given a complex space X, we shall define a new infinitesimal form [Formula: see text] of what is known as the Kobayashi pseudo-distance dx. At each point x of X, [Formula: see text] defines a pseudo-norm in the tangent space TxX; it satisfies the usual conditions for a norm except that [Formula: see text] may vanish for a nonzero vector υ. It turns out that the new pseudo-metric is the double dual of the old pseudo-metric Fx defined in [2] and [4]. This means that the indicatrix of [Formula: see text] is the convex hull of the indicatrix of Fx. In particular, [Formula: see text]. Advantages of [Formula: see text] over Fx are two-fold. First, [Formula: see text] satisfies the usual convexity condition, i.e., [Formula: see text]. (In [3] Lang calls Fx a semi-length function since it does not, in general, satisfy the convexity condition.) Second, it is defined on Zariski tangent spaces. It can be easily shown that [Formula: see text] is upper semicontinuous at nonsingular points of X. The upper semicontinuity for Fx is known also only in the nonsingular case. Although [Formula: see text], it can be shown, at least when X is nonsingular, that [Formula: see text] induces dx.

scholarly journals A maximum principle

1979 ◽  
Vol 28 (1) ◽  
pp. 23-26
Author(s):  
Kung-Fu Ng
Keyword(s):  
Maximum Principle ◽  
Extreme Point ◽  
Convex Space ◽  
Maximal Element ◽  
Locally Convex Space ◽  
Compact Set ◽  
Natural Manner ◽  
Upper Semicontinuous ◽  
Nonempty Compact ◽  
Locally Convex

AbstractLet K be a nonempty compact set in a Hausdorff locally convex space, and F a nonempty family of upper semicontinuous convex-like functions from K into [–∞, ∞). K is partially ordered by F in a natural manner. It is shown among other things that each isotone, upper semicontinuous and convex-like function g: K → [ – ∞, ∞) attains its K-maximum at some extreme point of K which is also a maximal element of K.Subject classification (Amer. Math. Soc. (MOS) 1970): primary 46 A 40.

An Efficient Improved Algorithm of Convex Hull

2014 ◽  
Vol 602-605 ◽  
pp. 3104-3106
Author(s):  
Shao Hua Liu ◽  
Jia Hua Zhang
Keyword(s):  
Convex Hull ◽  
Line Segment ◽  
Main Idea ◽  
Discrete Point ◽  
Directed Line ◽  
Generation Algorithm ◽  
Simple Program ◽  
Point Set ◽  
High Computational Efficiency ◽  
Improved Algorithm

This paper introduced points and directed line segment relation judgment method, the characteristics of generation and Graham method using the original convex hull generation algorithm of convex hull discrete points of the convex hull, an improved algorithm for planar discrete point set is proposed. The main idea is to use quadrilateral to divide planar discrete point set into five blocks, and then by judgment in addition to the four district quadrilateral internally within the point is in a convex edge. The result shows that the method is relatively simple program, high computational efficiency.

Sphericity Evaluation Based on Minimum Circumscribed Sphere Method

2012 ◽  
Vol 433-440 ◽  
pp. 3146-3151 ◽  
Author(s):  
Fan Wu Meng ◽  
Chun Guang Xu ◽  
Juan Hao ◽  
Ding Guo Xiao
Keyword(s):  
Convex Hull ◽  
Early Stage ◽  
Measured Data ◽  
Efficient Approach ◽  
Critical Data ◽  
Material Condition ◽  
Data Points ◽  
Point Set ◽  
Data Point

The search of sphericity evaluation is a time-consuming work. The minimum circumscribed sphere (MCS) is suitable for the sphere with the maximum material condition. An algorithm of sphericity evaluation based on the MCS is introduced. The MCS of a measured data point set is determined by a small number of critical data points according to geometric criteria. The vertices of the convex hull are the candidates of these critical data points. Two theorems are developed to solve the sphericity evaluation problems. The validated results show that the proposed strategy offers an effective way to identify the critical data points at the early stage of computation and gives an efficient approach to solve the sphericity problems.

scholarly journals Amount of failure of upper-semicontinuity of entropy in non-compact rank-one situations, and Hausdorff dimension

2015 ◽  
Vol 37 (2) ◽  
pp. 539-563 ◽  
Author(s):  
S. KADYROV ◽  
A. POHL
Keyword(s):  
Hausdorff Dimension ◽  
Homogeneous Spaces ◽  
Upper Semicontinuity ◽  
Metric Entropy ◽  
Real Rank ◽  
Semisimple Lie Group ◽  
Uniform Lattice ◽  
Finite Center ◽  
Rank One ◽  
Set Of Points

Recently, Einsiedler and the authors provided a bound in terms of escape of mass for the amount by which upper-semicontinuity for metric entropy fails for diagonal flows on homogeneous spaces $\unicode[STIX]{x1D6E4}\setminus G$, where $G$ is any connected semisimple Lie group of real rank one with finite center, and $\unicode[STIX]{x1D6E4}$ is any non-uniform lattice in $G$. We show that this bound is sharp, and apply the methods used to establish bounds for the Hausdorff dimension of the set of points that diverge on average.

scholarly journals A generalised quasi-variational inequality without upper semicontinuity

1996 ◽  
Vol 54 (2) ◽  
pp. 247-254 ◽  
Author(s):  
Paolo Cubiotti ◽  
Xian-Zhi Yuan
Keyword(s):  
Variational Inequality ◽  
Convex Subset ◽  
Existence Result ◽  
Upper Semicontinuity ◽  
Nonempty Closed Convex Subset ◽  
Positive Answer ◽  
Upper Semicontinuous ◽  
Quasi Variational Inequality ◽  
General Existence ◽  
Image Position

In this note we deal with the following problem: given a nonempty closed convex subset X of Rn and two multifunctions Γ : X → 2X and , to find ( such thatWe prove a very general existence result where neither Γ nor Φ are assumed to be upper semicontinuous. In particular, our result give a positive answer to an open problem raised by the first author recently.

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