The caratheodory number for thek-core

Thek-core of the setS ⁿ is the intersection of the convex hull of all setsA S with ¦SA¦<-k. The Caratheodory number of thek-core is the smallest integerf (d,k) with the property thatx core _{kS, S} ⁿ implies the existence of a subsetT S such thatx core_kT and ¦T¦f (d, k). In this paper various properties off(d, k) are established.Research of this author was partially supported by Hungarian National Science Foundation grant no. 1812.     

Finitely starlike sets whose F-stars have positive measure

Let and assume that there is a countable collection of lines {L _i : 1 i} such that (int cl S) and ((int cl S) S) L _i has one-dimensional Lebesgue measure zero, 1 i. Then every 4 point subset ofS sees viaS a set of positive two-dimensional Lebesgue measure if and only if every finite subset ofS sees viaS such a set. Furthermore, a parallel result holds with two-dimensional replaced by one-dimensional. Finally, setS is finitely starlike if and only if every 5 points ofS see viaS a common point. In each case, the number 4 or 5 is best possible.Supported in part by NSF grant DMS-8705336.     

ℒ2 sets which are almost starshaped

Let S be a subset of R ^d . The set S is said to be an set if and only if for every two points x and y of S, there exists some z S such that [x, z] [z, y] S. Clearly every starshaped set is an set, yet the converse is false and introduces an interesting question: Under what conditions will an set S be almost starshaped; that is, when will there exist a convex subset C of S such that every point of S sees some point of C via SThis paper provides one answer to the question above, and we have the following result: Let S be a closed planar set, S simply connected, and assume that the set Q of points of local nonconvexity of S is finite. If some point p of S see each member of Q via S, then there is a convex subset C of S such that every point of S sees some point of C via S.     

The hypermetric cone is polyhedral

The hypermetric coneH _n is the cone in the spaceR ^n(n–1)/2 of all vectorsd=(d _ij)_1i<jn satisfying the hypermetric inequalities: –_1ijn z _j z _j d _ij 0 for all integer vectorsz inZ ⁿ with –_1in z _i =1. We explore connections of the hypermetric cone with quadratic forms and the geometry of numbers (empty spheres andL-polytopes in lattices). As an application, we show that the hypermetric coneH _n is polyhedral.     

On the best estimate for perimeters of plane sets with the angle property

We investigate the problem of finding the maximum length of perimeters of plane sets with fixed diameter d, such that every point of the boundary of the set is a vertex of an open angle of opening which does not intersect the set. First we consider plane curves which satisfy such angle property in a finite number of directions, and among them we find the one of maximum length. Then we prove that the perimeter of any plane set with the angle property is less than or equal to d(sin /2)^-2; this is the best estimate when /2.     

A metric space of directions in Minkowski space

A new angular measure in a d-dimensional Minkowski space M was introduced recently. It determines the lengths of rectifiable curves in the (d - 1)-dimensional topological sphere S of all directions in M. Thus, a length structure appears on S. This results in the appropriate intrinsic metric in S. The paper deals with some properties of the length structure and the resulting metric space S. In particular, it shows that diam S 2 .     

Approximation under an arc length constraint

Let f C[a, b]. LetP be a subset ofC[a, b], L b – a be a given real number. We say thatp P is a best approximation tof fromP, with arc length constraintL, ifA[p] _b ^a [1 + (p(x)) ²]dx L andp – f q – f for allq P withA[q] L. represents an arbitrary norm onC[a, b]. The constraintA[p] L might be interpreted physically as a materials constraint.In this paper we consider the questions of existence, uniqueness and characterization of constrained best approximations. In addition a bound, independent of degree, is found for the arc length of a best unconstrained Chebyshev polynomial approximation.The work of L. L. Keener is supported by the National Research Council of Canada Grant A8755.     

Variations on the theme of repeated distances

We give an asymptotically sharp estimate for the error term of the maximum number of unit distances determined byn points in ^{d, d}4. We also give asymptotically tight upper bounds on the total number of occurrences of the favourite distances fromn points in ^{d, d}4. Related results are proved for distances determined byn disjoint compact convex sets in ².At the time this paper was written, both authors were visiting the Technion — Israel Institute of Technology.     

The cross-section body, plane sections of convex bodies and approximation of convex bodies, I

For a convex body K ^d we investigate three associated bodies, its intersection body IK (for 0int K), cross-section body CK, and projection body IIK, which satisfy IKCKIIK. Conversely we prove CKconst₁(d)I(K–x) for some xint K, and IIKconst₂ (d)CK, for certain constants, the first constant being sharp. We estimate the maximal k-volume of sections of 1/2(K+(-K)) with k-planes parallel to a fixed k-plane by the analogous quantity for K; our inequality is, if only k is fixed, sharp. For L ^d a convex body, we take n random segments in L, and consider their Minkowski average D. We prove that, for V(L) fixed, the supremum of V(D) (with also nN arbitrary) is minimal for L an ellipsoid. This result implies the Petty projection inequality about max V((IIM)*), for M ^d a convex body, with V(M) fixed. We compare the volumes of projections of convex bodies and the volumes of the projections of their sections, and, dually, the volumes of sections of convex bodies and the volumes of sections of their circumscribed cylinders. For fixed n, the pth moments of V(D) (1p<) also are minimized, for V(L) fixed, by the ellipsoids. For k=2, the supremum (nN arbitrary) and the pth moment (n fixed) of V(D) are maximized for example by triangles, and, for L centrally symmetric, for example by parallelograms. Last we discuss some examples for cross-section bodies.Research (partially) supported by Hungarian National Foundation for Scientific Research, Grant No. 41.     

On elementary equivalence and isomorphism of clone segments

The principal application of a general theorem proved here shows that for any choice 1mnp of integers there exist metric spacesX andY such that the initialk-segments of their clones of continuous maps coincide exactly whenkm, are isomorphic exactly whenkn, and are elementarily equivalent exactly whenkp.Dedicated to Prof. László Fuchs on the occasion of his 70th birthday     

Improved Krasnosel'skii theorems for the dimension of the kernel of a starshaped set

We will establish the following improved Krasnosel'skii theorems for the dimension of the kernel of a starshaped set: For each k and d, 0 k d, define f(d,k) = d+1 if k = 0 and f(d,k) = max{d+1,2d–2k+2} if 1 k d.Theorem 1. Let S be a compact, connected, locally starshaped set in R^d, S not convex. Then for a k with 0 k d, dim ker S k if and only if every f(d, k) lnc points of S are clearly visible from a common k-dimensional subset of S.Theorem 2. Let S be a nonempty compact set in R^d. Then for a k with 0 k d, dim ker S k if and only if every f (d, k) boundary points of S are clearly visible from a common k-dimensional subset of S. In each case, the number f(d, k) is best possible for every d and k.     

The steinitz lemma inl inf∞ sup2

Let (a, b) be a pair of non-negative numbers such that (1)a, b1 and (2)a+b3. Letu ₁,...,u _n be a sequence of vectors from the set {(x, y)R ²: |x|, |y|1}, withu ₁+...+u _n =0. It is shown that there is a permutation of indices such that all partial sumsu ₍₁₎+...+u _(k) lie in the rectangle |x|a, |y|b. Conditions (1) and (2) are also necessary.     

On low-dimensional faces that high-dimensional polytopes must have

We prove that every five-dimensional polytope has a two-dimensional face which is a triangle or a quadrilateral. We state and discuss the following conjecture: For every integerk1 there is an integer f(k) such that everyd-polytope,df(k), has ak-dimensional face which is either a simplex or combinatorially isomorphic to thek-dimensional cube.We give some related results concerning facet-forming polytopes and tilings. For example, sharpening a result of Schulte [25] we prove that there is no face to face tiling of ⁵ with crosspolytopes.Supported in part by a BSF Grant and by I.H.E.S, Bures-Sur-Yvette.     

Staircase kernels for orthogonald-polytopes

LetS be a finite union of boxes inR ^d . Forx inS, defineA _x ={yx is clearly visible fromy via staircase paths inS}, and let KerS denote the staircase kernel ofS. Then KerS={A _x x is a point of local nonconvexity ofS}. A similar result holds with clearly visible replaced by visible and points of local nonconvexity ofS replaced by boundary points ofS.Supported in part by NSF grant DMS-9207019.     

Reconstructing dimensions of intersections of convex sets

Denoting by dimA the dimension of the affine hull of the setA, we prove that if {K _i:i T} and {K _i ^j :i T} are two finite families of convex sets inR ⁿ and if dim {K _i :i S} = dim {K _i ^j :i S}for eachS T such that|S| n + 1 then dim {K _i :i T} = dim {K _i : {i T}}.     

On the Sobolev distance of convex bodies

Summary LetK ^d denote the cone of all convex bodies in the Euclidean spaceK ^d . The mappingK h _K of each bodyK K ^d onto its support function induces a metric _w onK ^d by" _w (K, L)h _L –h _{K w} where _w is the Sobolev I-norm on the unit sphere . We call _w (K, L) the Sobolev distance ofK andL. The goal of our paper is to develop some fundamental properties of the Sobolev distance.     

Ribbon tilings from spherical ones

The problem of classifying all tile-k-transitive tilings of the infinite 2-dimensional ribbon (and pinched-ribbon) is shown to be solvable by classifying certain tile-k-transitive tilings of the sphere, for all k. Complete results are listed for k3.Supported by the Deutsche Forschungsgemeinschaft.     

Weierstrass points and multiple intersection points of closed geodesics

We classify smooth complex projective surfaces of degreed and class , satisfying either (i) –d16, or (ii) 25. All these surfaces are rational or ruled. Indeed, we prove that the smallest value of the class of a non-ruled surface is 30 and in fact there are at least two surfacesS, both of degreed=10 and sectional genusg=6, with Kodaira dimension (S)=0 and class =30. Finally, we classify the smoothk-folds (k3) whose sectional surface has class 23.     

Distance-Regular Graphs with c i = b d-i and Antipodal Double Covers

Let be a distance-regular graph of diameter d and valency k > 2. Suppose there exists an integer s with d 2s such that c _i = b _d-i for all 1 i s. Then is an antipodal double cover.     

On a generalization of the Gallai-Sylvester theorem

It is shown that ifS ^d , affS=aff ^d , and every hyperplane spanned by (a subset of)S misses fewer thank points ofS(k2), then (a) #Skm ifd=2m–1 is odd and (b) #Skm+1 ifd=2m is even. We also fully describe the extreme sets for which equality holds in (a) or in (b). For oddd the proofs are later modified to purely algebraic ones, and carry over to , where is an arbitrary field. For evend, (b) is generally not true when , but we prove some weaker inequalities that do hold over arbitrary fields.This is part of a Ph.D. thesis, supervised by Professor Micha A. Perles at the Hebrew University of Jerusalem. This research was supported in part by the Landau Center for Mathematical Research.