A decomposition theorem form-convex sets

LetS be a closedm-convex subset of the plane,m≧2,Q the set of points of local nonconvexity ofS, with convQ ⊆S. If there is some pointp in [(bdryS) ∩ (kerS)] ∼Q, thenS is a union ofm−1 closed convex sets. The result is best possible for everym.     

On the intersection of maximalm-convex subsets

A subsetS of a real linear spaceE is said to bem-convex providedm≧2, there exist more thanm points inS, and for eachm distinct points ofS at least one of the ( ₂ ^m ) segments between thesem points is included inS. InE, letxy denote the segment between two pointsx andy. For any pointx inSυE, letS _x ={y: xyυS}. The kernel of a setS is then defined as {xεS: S _x=S}. It is shown that the kernel of a setS is always a subset of the intersection of all maximalm-convex subsets ofS. A sufficient condition is given for the intersection of all the maximalm-convex subsets of a setS to be the kernel ofS.     

On a problem of Erdős concerning property B

A family ℱ of sets has propertyB if there exists a setS such thatS∩F≠0 andS⊄F for everyF∈ℱ. ℱ has propertyB(s) if there exists a setS such that 0<|F∩S|<s for everyF∈ℱ. Denote bym(n) (respectivelym(n, s)) the size of a smallest family ofn-element sets not having propertyB (respectivelyB(s)). P. Erdős has asked whetherm(n, s)≧m (s) for alln≧s. We show that, in general, this inequality does not hold.     

General decomposition theorems form-convex sets in the plane

A setS inR ^dis said to bem-convex,m≧2, if and only if for everym distinct points inS, at least one of the line segments determined by these points lies inS. Clearly any union ofm?1 convex sets ism-convex, yet the converse is false and has inspired some interesting mathematical questions: Under what conditions will anm-convex set be decomposable intom?1 convex sets? And for everym≧2, does there exist aσ(m) such that everym-convex set is a union ofσ(m) convex sets? Pathological examples convince the reader to restrict his attention to closed sets of dimension≦3, and this paper provides answers to the questions above for closed subsets of the plane. IfS is a closedm-convex set in the plane,m ≧ 2, the first question may be answered in one way by the following result: If there is some lineH supportingS at a pointp in the kernel ofS, thenS is a union ofm ? 1 convex sets. Using this result, it is possible to prove several decomposition theorems forS under varying conditions. Finally, an answer to the second question is given: Ifm≧3, thenS is a union of (m?1)³2 ^m?3 or fewer convex sets.     

Inspheres and inner products

Among normed linear spacesX of dimension ≧3, finite-dimensional Hilbert spaces are characterized by the condition that for each convex bodyC inX and each ballB of maximum radius contained inC,B’s center is a convex combination of points ofB ∩ (boundary ofC). Among reflexive Banach spaces of dimension ≧3, general Hilbert spaces are characterized by a related but weaker condition on inscribed balls. Research of the first author was partially supported by the U.S. National Science Foundation. Research of the second and third authors was supported by the Consiglio Nazionale delle Ricerche and the Ministero della Pubblica Istruzione of Italy, while they were visiting the University of Washington, Seattle, USA.     

On visibility and covering by convex sets

A setX⊆ℝ ^d isn-convex if among anyn of its points there exist two such that the segment connecting them is contained inX. Perles and Shelah have shown that any closed (n+1)-convex set in the plane is the union of at mostn ⁶ convex sets. We improve their bound to 18n ³, and show a lower bound of order Ω(n ²). We also show that ifX⊆ℝ² is ann-convex set such that its complement has λ one-point path-connectivity components, λ<∞, thenX is the union ofO(n ⁴+n ²λ) convex sets. Two other results onn-convex sets are stated in the introduction (Corollary 1.2 and Proposition 1.4). Research supported by Charles University grants GAUK 99/158 and 99/159, and by U.S.-Czechoslovak Science and Technology Program Grant No. 94051. Part of the work by J. Matoušek was done during the author’s visits at Tel Aviv University and The Hebrew University of Jerusalem. Part of the work by P. Valtr was done during his visit at the University of Cambridge supported by EC Network DIMANET/PECO Contract No. ERBCIPDCT 94-0623.     

B-convexity and reflexivity

If ε>1/4 andX is 3,ε-convex thenX is reflexive. Some additional values ofk and ε with k≧4 are found for whichk,ε-convexity implies reflexivity.     

Colouring Arcwise Connected Sets in the Plane I

Let ? be the family of finite collections ? where ? is a collection of bounded, arcwise connected sets in ℝ² which for any S, T∈? where S∩T≠∅, it holds that S∩T is arcwise connected. We investigate the problem of bounding the chromatic number of the intersection graph G of a collection ?∈?.  Assuming G is triangle-free, suppose there exists a closed Jordan curve C⊂ℝ² such that C intersects all sets of ? and for all S∈?, the following holds: (i) S∩(C∪int (C)) is arcwise connected or S∩int (C)=∅. (ii) S∩(C∪ext (C)) is arcwise connected or S∩ext (C)=∅.  Here int(C) and ext (C) denote the regions in the interior, resp. exterior, of C. Such being the case, we shall show that χ(?) is bounded by a constant independent of ?. Revised: December 3, 1998     

NON-EMPTY CROSS-2-INTERSECTING FAMILIES OF SUBSETS

Let Cdenote the set of all k-subests of an n-set.Assume Alohtain in Ca,and A lohtain in (A,B) is called a cross-2-intersecting family if |A B≥2 for and A∈A,B∈B.In this paper,the best upper bounds of the cardinalities for non-empty cross-2-intersecting familles of a-and b-subsets are obtained for some a and b,A new proof for a Frankl-Tokushige theorem[6] is also given.     

Primitive permutation groups of simple diagonal type

LetG be a finite primitive group such that there is only one minimal normal subgroupM inG, thisM is nonabelian and nonsimple, and a maximal normal subgroup ofM is regular. Further, letH be a point stabilizer inG. ThenH∩M is a (nonabelian simple) common complement inM to all the maximal normal subgroups ofM, and there is a natural identification ofM with a direct powerT ^m of a nonabelian simple groupT in whichH∩M becomes the “diagonal” subgroup ofT ^m: this is the origin of the title. It is proved here that two abstractly isomorphic primitive groups of this type are permutationally isomorphic if (and obviously only if) their point stabilizers are abstractly isomorphic. GivenT ^m, consider first the set of all permutational isomorphism classes of those primitive groups of this type whose minimal normal subgroups are abstractly isomorphic toT ^m. Secondly, form the direct productS _m×OutT of the symmetric group of degreem and the outer automorphism group ofT (so OutT=AutT/InnT), and consider the set of the conjugacy classes of those subgroups inS _m×OutT whose projections inS _m are primitive. The second result of the paper is that there is a bijection between these two sets. The third issue discussed concerns the number of distinct permutational isomorphism classes of groups of this type, which can fall into a single abstract isomorphism class.     

Convex decompositions in the plane and continuous pair colorings of the irrationals

We address the structure of nonconvex closed subsets of the Euclidean plane. A closed subsetS⊆ℝ² which is not presentable as a countable union of convex sets satisfies the following dichotomy:

On characteristic properties of singular operators

For a linear operatorS in a Hilbert space ℋ, the relationship between the following properties is investigated: (i)S is singular (= nowhere closable), (ii) the set kerS is dense in ℋ, and (iii)D(S)∩ℛ(S)={0}.     

p-Congruences onp-regular semigroups

A pair (S, P) of a regular semigroupsS and a subsetP ofE _s whereE _s is the set of all idempotent elements ofS is called aP-regular semigroupS(P) if it satisfies the following:

Central localization and Gelfand-Kirillov dimension

LetR be a factor ring of the enveloping algebra of a finite dimensional Lie algebra over a fieldk. If the centre ofR, Z, consists of non-zero divisors inR, the ringR _z obtained by localizing at the non-zero elements ofZ becomes a finitely generated algebra over the fieldK which arises as the field of fractions ofZ. The Gelfand-Kirillov dimension of anR-moduleM is denotedd(M). In this paper it is shown that ifR _Z ⊗ _R M ≠ 0 thend(M) ≧d(R _Z ⊗ _R M) + tr. deg _k Z, whered (R _z ⊗M) is the Gelfand-Kirillov dimension ofR _z ⊗M) viewed as anR _z -module andR _z is viewed as a finitely generatedK-algebra (not as ak-algebra). The result is primarily of a technical nature.     

Decomposition of ranges of vector measures

The following conditions on a zonoidZ, i.e., a range of a non-atomic vector measure, are equivalent: (i) the extreme set containing 0 in its relative interior is a parallelepiped; (ii) the zonoidZ determines them-range of any non-atomic vector measure with rangeZ, where them-range of a vector measure μ is the set ofm-tuples (μ(S ₁), …, μ(S _m), whereS ₁, …S _m are disjoint measurable sets and (iii) there is avector measure space (X, Σ, μ) such that any finite factorization ofZ, Z =ΣZ _i , in the class of zonoids could be achieved by decomposing (X, Σ). In the case of ranges of non-atomic probability measures (i) is automatically satisfied, so (ii) and (iii) hold. Partially supported by NSF grant MCS-79-06634     

A characterization ofC 1-convex sets in Sobolev spaces

In this paper we prove that, ifK is a closed subset ofW ₀ ^1,p (Ω,R ^m ) with 1<p<+∞ andm≥1, thenK is stable under convex combinations withC ¹ coefficients if and only if there exists a closed and convex valued multifunction from Ω toR ^m such that The casem=1 has already been studied by using truncation arguments which rely on the order structure ofR (see [2]). In the casem>1 a different approach is needed, and new techniques involving suitable Lipschitz projections onto convex sets are developed. Our results are used to prove the stability, with respect to the convergence in the sense of Mosco, of the class of convex sets of the form (*); this property may be useful in the study of the limit behaviour of a sequence of variational problems of obstacle type. This article was processed by the author using the Springer-Verlag TEX mamath macro package 1990     

Cyclotomic units in z p -extensions

LetK ₀ be the maximal real subfield of the field generated by thep-th root of 1 over ℚ, andK∞ be the basic Z_p-extension ofK ₀ for a fixed odd primep. LetK _n be itsn-th layer of this tower. For eachn, we denote the Sylowp-subgroup of the ideal class group ofK _n byA _n , and that ofE _n C _n byB _n , whereE _n (resp.C _n ) is the group of units (resp. cyclotomic units ofK _n . In section 2 of this paper, we describe structures of the direct and inverse limits ofB _n . The direct limit, in particular, is shown to be a direct sum of λ copies ofp-divisible groups and a finite group M, where λ is the Iwasawa λ-invariant for K∞ overK ₀. In section 3, we prove that the capitulation ofA _n inA _m is isomorphic to M form ≫n ≫ 0 by using cohomological arguments. Hence if we assume Greenberg’s conjecture (λ = 0), thenA _n is isomorphic toB _n forn ≫ 0. This paper was supported in part by a research fund for junior scholars, Korea Research Foundation The present studies were supported in part by the Basic Science Research Institute program, Ministry of Education, 1989.     

On invariant additive subgroups

Suppose thatR is a prime ring with the centerZ and the extended centroidC. An additive subgroupA ofR is said to be invariant under special automorphisms if (1+t)A(1+t)⁻¹ ⊆A for allt ∈R such thatt ²=0. Assume thatR possesses nontrivial idempotents. We prove: (1) If chR ≠ 2 or ifRC ≠C ₂, then any noncentral additive subgroup ofR invariant under special automorphisms contains a noncentral Lie ideal. (2) If chR=2,RC=C ₂ andC ≠ {0, 1}, then the following two conditions are equivalent: (i) any noncentral additive subgroup invariant under special automorphisms contains a noncentral Lie ideal; (ii) there isα ∈Z / {0} such thatα ² Z ⊆ {β ²:β ∈Z}.     

Banach spaces with a weak cotype 2 property

We study the Banach spacesX with the following property: there is a numberδ in ]0,1[ such that for some constantC, any finite dimensional subspaceE ⊂X contains a subspaceF ⊂E with dimF≧δ dimE which isC-isomorphic to a Euclidean space. We show that if this holds for someδ in ]0,1[ then it also holds for allδ in ]0,1[ and we estimate the functionC=C(δ). We show that this property holds iff the “volume ratio” of the finite dimensional subspaces ofX are uniformly bounded. We also show that (althoughX can have this property without being of cotype 2)L ₂(X) possesses this property iffX if of cotype 2. In the last part of the paper, we study theK-convex spaces which have a dual with the above property and we relate it to a certain extension property.     

Large centralizers in finite solvable groups

In 1955 R. Brauer and K. A. Fowler showed that ifG is a group of even order >2, and the order |Z(G)| of the center ofG is odd, then there exists a strongly real) elementx∈G−Z whose centralizer satisfies|C _G(x)|>|G|^1/3. In Theorem 1 we show that every non-abeliansolvable groupG contains an elementx∈G−Z such that|C _G(x)|>[G:G′∩Z]^1/2 (and thus|C _G(x)|>|G|^1/3). We also note that if non-abelianG is either metabelian, nilpotent or (more generally) supersolvable, or anA-group, or any Frobenius group, then|C _G(x)|>|G|^1/2 for somex∈G−Z. In Theorem 2 we prove that every non-abelian groupG of orderp ^mqⁿ (p, q primes) contains a proper centralizer of order >|G|^1/2. Finally, in Theorem 3 we show that theaverage |C(x)|, x∈G, is ≧c|G| ^1/3 for metabelian groups, wherec is constant and the exponent 1/3 is best possible.