Open Half-spaces Research Articles

Geometric methods in the study of irregularities of distribution

Letν be a signed measure on E d with νE d =0 and ¦ν¦Ed<∞. DefineD s(ν) as sup ¦νH¦ whereH is an open halfspace. Using integral and metric geometric techniques results are proved which imply theorems such as the following.Theorem A. Letν be supported by a finite pointsetp i. ThenD s(ν)>c d(δ1/δ 2)1/2{∑ i(νp i)2}1/2 whereδ 1 is the minimum distance between two distinctp i, andδ 2 is the maximum distance. The numberc d is an absolute dimensional constant. (The number .05 can be chosen forc 2 in Theorem A.)Theorem B. LetD be a disk of unit area in the planeE 2, andp 1,p 2,...,p n be a set of points lying inD. If m if the usual area measure restricted toD, while γnP i=1/n defines an atomic measure γn, then independently of γn,nD s(m −γ n)≥ .0335n 1/4. Theorem B gives an improved solution to the Roth “disk segment problem” as described by Beck and Chen. Recent work by Beck shows thatnD s(m −γ n)≥cn 1/4(logn)−7/2.

The number of extreme pairs of finite point-sets in Euclidean spaces

To points p and q of a finite set S in d-dimensional Euclidean space E d are extreme if { p, q} = S ∩ h, for some open halfspace h. Let e 2 ( d) ( n) be the maximum number of extreme pairs realized by any n points in E d . We give geometric proofs of e 2 (2)(n) = ⌊ 3n 2 ⌋ , if n⩾4, and e 2 (3)( n) = 3 n−6, if n⩾6. These results settle the question since all other cases are trivial.

Heating of a fully saturated darcian half-space: Pressure generation, fluid expulsion, and phase change

Analytical solutions are developed for the pressurization, expansion, and flow of one- and two-phase liquids during heating of fully saturated and hydraulically open Darcian half-spaces subjected to a step rise in temperature at its surface. For silicate materials, advective transfer is commonly unimportant in the liquid region; this is not always the case in the vapor region. Volume change is commonly more important than heat of vaporization in determining the position of the liquid-vapor interface, assuring that the temperatures cannot be determined independently of pressures. Pressure increases reach a maximum near the leading edge of the thermal front and penetrate well into the isothermal region of the body. Mass flux is insensitive to the hydraulic properties of the half-space.

Remarks on piecewise-linear algebra

This note studies some of the basic properties of the category whose objects are finite unions of (open and closed) and whose morphisms are (not necessarily continu- ous) piecewise-linear maps. Introduction* A function f: V —>W between real vector spaces is piecewise-linear (PL) if there exists a partition of V into polyhedra X€ (i.e., relative interiors of polyhedra) such that / is affine on each Xt. (As distinct to the case of PL-topology, no continuity is required of /.) Images and preimages under PL-maps give rise to finite unions of open polyhedra, or PL-seίs; conversely PL functions can be characterized by the fact that their graphs are PL-sets. This paper studies some basic algebraic properties of the category PL, proving in particular that it is an exact category, and in fact a pretopos. A classification is given for the isomorphism classes of objects of PL, in terms of a two-generator semiring. The first section recalls without proof some facts from polyhedral geometry needed in the paper. Except for the setting of a unified notation and for minor generalizations, the material there is well known. The second section defines PL maps and sets, and studies the category. The main results (leading to the classification theorem) are given in the last section. 1* Review of polyhedral geometry* The following conventions and definitions hold throughout. All vector spaces are finite- dimensional spaces over the reals R; a flat means an affine sub- manifold of some such space V, and the closed half-spaces associated to a linear /: V —> R and an r in R (or associated to the hyperplane {x\f(x) = r},) are the sets {x\f(x) ^ r} and {x\f(x) ^ r}. The cor- responding open half-spaces are obtained by using strict inequalities in the above. A half-line (closed or open) is the intersection of a line L in V with a (closed or open) half-space not containing L.

Comments on "Transmission from a Rectangular Waveguide into Half Space through a Rectangular Aperture" [Letters

The authors of the above computer program description failed to acknowledge several publications dedicated to the same subject. An open-ended waveguide terminated by a flat infinite metal flange and radiating into an open half space of homogeneous, isotropic, and possibly Iossy medium was considered in [1]. The electromagnetic field distribution, the equivalent admittance of the aperture, and the reflection coefficient were presented. In another publication [2], the same approach was applied to radiation through a partial rectangular aperture; it was then also extended to radiation into a slab. The study of a rectangular cavity made from a section of open-ended rectangular waveguide was also treated and further developed in a third publication [3]. Computer programs have been available for all these problems on a complimentary basis and were widely distributed.

Decomposition theorems for 3-convex subsets of the plane

Let S be a 3-convex subset of the plane. If (cl S ~ £>) S int (cl S) or if (cl S ~ S) g bdry (cl S), then S is expressible as a union of four or fewer convex sets. Otherwise, S is a union of six or fewer convex sets. In each case, the bound is best possible. 1* Introduction. Let S be a subset of Rd. Then S is said to be 3-convex iff for every three distinct points in S, at least one of the segments determined by these points lies in S. Valentine [2] has proved that for S a closed, 3-convex subset of the plane, S is expressible as a union of three or fewer closed convex sets. We are interested in obtaining a similar decomposition without requiring the set S to be closed. The following definitions and results obtained by Valentine will be useful. For S S Rd> a point x in S is a point of local convexity of S iff there is some neighborhood U of x such that, if y, zeSCiU, then [y, A S S. If S fails to be locally convex at some point q in S, then q is called a point of local nonconvexity (lnc point) of S. Let S be a closed, connected, 3-convex subset of the plane, and let Q denote the closure of the set of isolated lnc points of S. Valentine has proved that for S not convex, then card Q ^ 1, Q lies in the convex kernel of S, and Q § bdry (conv Q). An edge of bdry (conv Q) is a closed segment (or ray) in bdry (conv Q) whose endpoints are in Q. We define a leaf of S in the following manner: In case card Q ^ 3, let L be the line determined by an edge of bdry (conv Q), Llf L2 the corresponding open half spaces. Then L supports conv Q, and we may assume conv Q § cl (LJ. We define W — cl (L2 f] S) to be a leaf of S. For 2 ^ card Q ^ 1, constructions used by Valentine may be employed to decompose S into two closed convex sets, and we define each of these convex sets to be a leaf of S. By Valentine's results, every point of S is either in conv Q or in some leaf W of S (or both), and every leaf W is convex. Moreover, Valentine obtains his decomposition of S by showing that for any collection {sj of disjoint edges of bdry (conv Q), with {Wt} the corresponding collection of leaves, conv Q U (U Wt) is closed and convex. Finally, we will use the following familiar definitions: For x, y in S, we say x see y via S iff the corresponding segment [x, y] lies in S. A subset T of S is visually independent via S iff for every

On closed additive semigroups in $E\sp{n}$

Let C(A) be the closed additive semigroup generated by a set A z En. A simple necessary and sufficient condition on A for C(A) to be a group is derived. An example which arose in the theory of random walks and stimulated these purely geometrical considerations is discussed at the end. For an arbitrary subset A of the n-dimensional Euclidean space El (n>1) let S(A) denote the smallest additive semigroup containing A, Athe closure of A with respect to the Euclidean topology, and C(A):= (S(A))-. Obviously, C(A) is the smallest closed additive semigroup containing A, i.e., the closed additive semigroup generated by A. DEFINITION. A c En is called omnilateral, if A $ 0 and for every hyperplane H through the origin with A t H there are points of A in each of the two open half-spaces produced by H. In other words, A is omnilateral if A $ 0 and u e En, x e A and ux> 0 imply the existence of an x* E A with ux* <0. Some immediate consequences are listed in the following lemma. LEMMA. (1) A is omnilateral if and only if S(A) is omnilateral. (2) A is omnilateral if and only if Ais omnilateral. (3) An additive group A is always omnilateral. (4) The image of an omnilateral set AC E under a linear transformation from En to Em is an omnilateral set in Em. We say a set A c En is genuinely n-dimensional if there is no hyperplane through the origin 0 containing A. A0 denotes the convex hull of A. THEOREM 1. A genuinely n-dimensional set A is omnilateral if and only fO is an inner point of Ac. PROOF. The if-part is trivial. Suppose, on the other hand, 0 is an exterior or a boundary point of Ac. There would be a hyperplane through Received by the editors August 31, 1971. AMS 1969 subject classifications. Primary 20M99, 50B99; Secondary lOE05, 52A20, 60J15.

On tight immersions of maximal codimension

Let f : M--+ be an immersion of a compact differentiable manifold of dimension n into a Euclidean space of dimension m. The immersion f is called tight if there exists no immersion of with smaller total Lipschitz-Killing curvature [3] in any Euclidean space. The immersion f is called substantial if f(M) is not contained in any hyperplane of E =. Kuiper [4] has shown that if f is both tight and substantial, then m<=N= 89 In case m=N and n = 2 (so that N=5) , he has shown that i f f is tight and substantial then M z must be diffeomorphic to the real projective plane and f must be an embedding onto a real algebraic variety, in fact onto a Veronese surface. In this paper we prove the corresponding result in higher dimensions. Our hypothesis is, in fact, weaker. The immersion f is said to have the two-piece property if every hyperplane divides it into at most two pieces, or more exactly, if for every hyperplane HOE, f l ( H 0 and f l ( H 2 ) are both connected sets, where H~ and H 2 a r e the two open half-spaces which make up the complement of H in E'. A tight immersion has the two-piece property, but not necessarily conversely (cf. [-6,9]). However, for the case of curves and surfaces the two properties are equivalent. Let A be a real vector space of dimension n + 1 and consider the map v~-, v| from A to A| Take a metric in A and restrict the map to the unit sphere centered at the origin. Since ( -v ) |174 this map takes each pair of antipodal points to the same point. Hence it induces a map of the real projective n-space into A | A. As we shall see, this last map is an embedding, and the image Vlies substantially in an affine subspace of dimension N = 89 n (n + 3). W e call any submanifold projectively equivalent to V and lying in an affine or projective space a Veronese n-rnan~)ld. Any Veronese manifold is tightly embedded [4]. We can now state our main result.

Weak Compactness and Separation

The purpose of this paper is to develop characterizations of weakly compact subsets of a Banach space in terms of separation properties. The sets A and B are said to be separated by a hyperplane H if A is contained in one of the two closed half-spaces determined by H, and B is contained in the other; A and B are strictly separated by H if A is contained in one of the two open half-spaces determined by H, and B is contained in the other. The following are known to be true for locally convex topological linear spaces.