A Peak Point Theorem for Uniform Algebras Generated by Smooth Functions on Two-Manifolds

We establish the peak point conjecture for uniform algebrasgenerated by smooth functions on two-manifolds: if A is a uniformalgebra generated by smooth functions on a compact smooth two-manifoldM, such that the maximal ideal space of A is M, and every pointof M is a peak point for A, then A = C(M). We also give an alternativeproof in the case when the algebra A is the uniform closureP(M) of the polynomials on a polynomially convex smooth two-manifoldM lying in a strictly pseudoconvex hypersurface in Cⁿ.     

Failure of Polynomial Approximation on Polynomially Convex Subsets of the Sphere

An example is presented of a compact polynomially convex subsetof the unit sphere in C³ on which the polynomials in the complexcoordinate functions are not dense in the continuous functions.Also presented is an example of a compact polynomially convexsmooth solid torus lying in the unit sphere in C⁵ with the samefailure of polynomial approximation.     

An O(n2) algorithm for a controllable machine scheduling problem

A single-machine scheduling problem with controllable processingtimes is discussed in this paper. For some jobs, the processingtime can be crashed up to u units of time with the additionalcost c per unit of time crashed. The object is to find an optimalprocessing sequence as well as crash activities to minimizetotal costs of completion and crash. This problem is shown tobe polynomially solvable, and an O(n²) algorithm is given togetherwith the theoretical proof.     

De Paepe's Disc has Nontrivial Polynomial Hull

The topological disc (De Paepe's) isshown here to have non-trivial polynomially convex hull. Infact, the authors show that this holds for all discs of theform , where f is holomorphicon |z|r, and f(z=z²+a₃z³+..., with all coefficients a_n real,and at least one a_2n+1 0. 2000 Mathematics Subject Classification32E20.     

Zero-Sets of Quaternionic and Octonionic Analytic Functions with Central Coefficients

We prove that the zero set of any quaternionic (or octonionic)analytic function f with central (that is, real) coefficientsis the disjoint union of codimension two spheres in R⁴ or R⁸(respectively) and certain purely real points. In particular,for polynomials with real coefficients, the complete root-setis geometrically characterisable from the lay-out of the rootsin the complex plane. The root-set becomes the union of a finitenumber of codimension 2 Euclidean spheres together with a finitenumber of real points. We also find the preimages f^–1for any quaternion (or octonion) A. We demonstrate that this surprising phenomenon of complete spheresbeing part of the solution set is very markedly a special ‘real’phenomenon. For example, the quaternionic or octonionic Nthroots of any non-real quaternion (respectively octonion) turnout to be precisely N distinct points. All this allows us todo some interesting topology for self-maps of spheres.     

Optimal Smoothing for Convex Polytopes

It is proved that, given a convex polytope P in Rⁿ, togetherwith a collection of compact convex subsets in the interiorof each facet of P, there exists a smooth convex body arbitrarilyclose to P that coincides with each facet precisely along theprescribed sets, and has positive curvature elsewhere. 2000Mathematics Subject Classification 53A07, 52B11, 53C45.     

Convolutions, Transforms, and Convex Bodies

The paper studies convex bodies and star bodies in Rⁿ by usingRadon transforms on Grassmann manifolds, p-cosine transformson the unit sphere, and convolutions on the rotation group ofRⁿ. It presents dual mixed volume characterizations of i-intersectionbodies and L_p-balls which are related to certain volume inequalitiesfor cross sections of convex bodies. It considers approximationsof special convex bodies by analytic bodies and various finitesums of ellipsoids which preserve special geometric properties.Convolution techniques are used to derive formulas for mixedvolumes, mixed surface measures, and p-cosine transforms. Theyare also used to prove characterizations of geometric functionals,such as surface area and dual quermassintegrals. 1991 MathematicsSubject Classification: 52A20, 52A40.     

On the Least Q-order of Convergence of Variable Metric Algorithms

It is shown in this paper that the infimum of the Q-order ofthe convergence of variable metric algorithms is only 1, eventhough the objective function is twice continuously differentiableand uniformly convex. It is shown by example that the Q-ordercan be 1 + 1/N for any large N, though the R-order is (1+N)^1/2.     

Matheron's Conjecture for the Covariogram Problem

The covariogram of a convex body K provides the volumes of theintersections of K with all its possible translates. Matheronconjectured in 1986 that this information determines K amongall convex bodies, up to certain known ambiguities. It is provedthat this is the case if K R² is not C¹, or it is not strictlyconvex, or its boundary contains two arbitrarily small C² openportions ‘on opposite sides’. Examples are alsoconstructed that show that this conjecture is false in Rⁿ forany n 4.     

Points of {varepsilon} Differentiability of Lipschitz Functions from Rn to Rn-1

This paper proves that for every Lipschitz function f : RⁿR^m,m < n, there exists at least one point of -differentiabilityof f which is in the union of all m-dimensional affine subspacesof the form q0 +span{q1,q2,...,q_m}, where q_j (j = 0,1,...,m)are points in Rⁿ with rational coordinates. 2000 MathematicsSubject Classification 26B05, 26B35.     

On Closed Sets with Convex Projections

A shadow of a subset A of Rⁿ is the image of A under a projectiononto a hyperplane. Let C be a closed nonconvex set in Rⁿ suchthat the closures of all its shadows are convex. If, moreover,there are n independent directions such that the closures ofthe shadows of C in those directions are proper subsets of therespective hyperplanes then it is shown that C contains a copyof R^n–2. Also for every closed convex set B ‘minimalimitations’ C of B are constructed, that is, closed subsetsC of B that have the same shadows as B and that are minimalwith respect to dimension.     

A Volume Inequality Concerning Sections of Convex Sets

We give a lower estimate of the volume of a compact convex subsetof Rⁿ in terms of the volumes of its sections by n pairwiseorthogonal affine hyperplanes.     

Successive Determination and Verification of Polytopes by their X-Rays

It is shown that each convex polytope P in ^d can be verifiedby ([d/(d–k)] + 1) k-dimensional X-rays. This means thatP is uniquely determined by these X-rays and the choice of thedirection of each X-ray depends only on P. Examples are constructedto show that in general this number cannot be reduced. Further,it is shown that each convex polytope P in ³ can be successivelydetermined by only two one-dimensional X-rays. This means thatP is uniquely determined by one X-ray taken in an arbitrarydirection together with another whose direction depends onlyon the first X-ray. The results extend those for the case d= 2 of Giering and of Edelsbrunner and Skiena.     

An Euler-Type Version of the Local Steiner Formula for Convex Bodies

The purpose of this note is to establish a new version of thelocal Steiner formula and to give an application to convex bodiesof constant width. This variant of the Steiner formula generalizesresults of Hann [3] and Hug [6], who use much less elementarytechniques than the methods of this paper. In fact, Hann askedfor a simpler proof of these results [4, Problem 2, p. 900].We remark that our formula can be considered as a Euclideananalogue of a spherical result proved in [2, p. 46], and thatour method can also be applied in hyperbolic space. For some remarks on related formulas in certain two-dimensionalMinkowski spaces, see Hann [5, p. 363]. For further information about the notions used below, we referto Schneider's book [9]. Let Kⁿ be the set of all convex bodiesin Euclidean space Rⁿ, that is, the set of all compact, convex,non-empty subsets of Rⁿ. Let S^n–1 be the unit sphere.For KKⁿ, let NorK be the set of all support elements of K, thatis, the pairs (x, u)RⁿxS^n–1 such that x is a boundarypoint of K and u is an outer unit normal vector of K at thepoint x. The support measures (or generalized curvature measures)of K, denoted by ₀(K.), ..., _n–1(K.), are the unique Borelmeasures on RⁿxS^n–1 that are concentrated on NorK andsatisfy [formula] for all integrable functions f:RⁿR; here denotes the Lebesguemeasure on Rⁿ. Equation (1), which is a consequence and a slightgeneralization of Theorem 4.2.1 in Schneider [9], is calledthe local Steiner formula. Our main result is the following.1991 Mathematics Subject Classification 52A20, 52A38, 52A55.     

Convex Bodies, Graphs and Partial Orders

A convex corner is a compact convex down-set of full dimensionin Rⁿ. Convex corners arise in graph theory, for instance asstable set polytopes of graphs. They are also natural objects of study in geometry, as they correspond to 1-unconditionalnorms in an obvious way. In this paper, we study a parameterof convex corners, which we call the content, that is relatedto the volume. This parameter has appeared implicitly before:both in geometry, chiefly in a paper of Meyer (Israel J. Math.} 55 (1986) 317–327) effectively using content to givea proof of Saint-Raymond's Inequality on the volume product of a convex corner, and in combinatorics, especially in apaper of Sidorenko (Order} 8 (1991) 331–340) relatingcontent to the number of linear extensions of a partial order.One of our main aims is to expose connections between workin these two areas. We prove many new results, giving in particular various generalizations of Saint-Raymond's Inequality. Contentalso behaves well under the operation of pointwise product oftwo convex corners; our results enable us to give counter-examplesto two conjectures of Bollobás and Leader Oper. TheoryAdv. Appl. 77 (1995) 13–24) on pointwise products. 1991Mathematics Subject Classification: 52C07, 51M25, 52B11, 05C60,06A07.     

From Trace States to States on the K0 Group of a Simple C*-Algebra

It is shown that any continuous affine surjection from a metrizableChoquet simplex onto a compact convex set occurs as the restrictionmap from the tracial state space onto the state space of theK₀ group of a separable unital simple C^*-algebra which is theinductive limit of a sequence of subhomogeneous C^*-algebras     

A Lie Theoretic Approach to Thompson's Theorems on Singular Values-Diagonal Elements and Some Related Results

Thompson's famous theorems on singular values–diagonalelements of the orbit of an nxn matrix A under the action (1)U(n) U(n) where A is complex, (2) SO(n) SO(n), where A isreal, (3) O(n) O(n) where A is real are fully examined. Coupledwith Kostant's result, the real semi-simple Lie algebra so_n,n yields (2) and hence (3) and the sufficient part (the hardpart) of (1). In other words, the curious subtracted term(s)are well explained. Although the diagonal elements correspondingto (1) do not form a convex set in Cⁿ, the projection of thediagonal elements into Rⁿ (or iRⁿ) is convex and the characterizationof the projection is related to weak majorization. An elementaryproof is given for this hidden convexity result. Equivalentstatements in terms of the Hadamard product are also given.The real simple Lie algebra su_{n, n} shows that such a convexityresult fits into the framework of Kostant's result. Convexityproperties and torus relations are studied. Thompson's resultson the convex hull of matrices (complex or real) with prescribedsingular values, as well as Hermitian matrices (real symmetricmatrices) with prescribed eigenvalues, are generalized in thecontext of Lie theory. Also considered are the real simple Liealgebras so_{p, q} and so_{p, q}, p < q, which yield the rectangularcases. It is proved that the real part and the imaginary partof the diagonal elements of complex symmetric matrices withprescribed singular values are identical to a convex set inRⁿ and the characterization is related to weak majorization.The convex hull of complex symmetric matrices and the convexhull of complex skew symmetric matrices with prescribed singularvalues are given. Some questions are asked.     

Isometries of the Space of Convex Bodies in Euclidean Space

The isometries of the space of convex bodies of E^d with respectto the Hausdorff-metric are precisely the mappings of the formC i(C) + D where i is a rigid motion of E^d and D a fixed convexbody.     

Viscoelastic boundary stabilization for a transmission problem in elasticity

We consider an anisotropic body which is constituted of twodifferent types of materials supporting a memory boundary conditionand we show that its energy decays uniformly as time goes toinfinity with the same rate as the relaxation function g, thatis, the energy decays exponentially when g decays exponentially,and polynomially when g decays polynomially.     

Plane with A{infty}-Weighted Metric not Bilipschitz Embeddable to Rn

A planar set G R² is constructed that is bilipschitz equivalentto (G,d^z), where (G, d) is not bilipschitz embeddable to anyuniformly convex Banach space. Here, Z (0, 1) and d^z denotesthe zth power of the metric d. This proves the existence ofa strong A weight in R², such that the corresponding deformedgeometry admits no bilipschitz mappings to any uniformly convexBanach space. Such a weight cannot be comparable to the Jacobianof a quasiconformal self-mapping of R². 2000 Mathematics SubjectClassification 54E40 (primary); 30C62, 30C65, 28A80 (secondary).