Convex mappings on some Reinhardt domains

In this paper, we consider the following Reinhardt domains. Let M = （M1, M2,..., Mn） ： [0,1] → [0,1]^n be a C2-function and Mj（0） = 0, Mj（1） = 1, Mj″ 〉 0, C1jr^pj-1 〈 Mj′（r） 〈 C2jr^pj-1, r∈ （0, 1）, pj 〉 2, 1 ≤ j ≤ n, 0 〈 C1j 〈 C2j be constants. Define
DM={z=（z1,z2,…,Zn）^T∈C^n：n∑j=1 Mj（｜zj｜）〈1}
Then DM C^n is a convex Reinhardt domain. We give an extension theorem for a normalized biholomorphic convex mapping f ： DM -→ C^n.     

Mean convex hulls and least area disks spanning extreme curves

We show that for any extreme curve in a 3-manifold M, there exist a canonical mean convex hull containing all least area disks spanning the curve. Similar result is true for asymptotic case in such that for any asymptotic curve , there is a canonical mean convex hull containing all minimal planes spanning Γ. Applying this to quasi-Fuchsian manifolds, we show that for any quasi-Fuchsian manifold, there exist a canonical mean convex core capturing all essential minimal surfaces. On the other hand, we also show that for a generic C³-smooth curve in the boundary of C³-smooth mean convex domain in ℝ³, there exist a unique least area disk spanning the curve.     

The Hahn–Banach Theorem for Partially Ordered Totally Convex, Positively Convex and Superconvex Modules

The Hahn–Banach Theorem for partially ordered totally convex modules [3] and a necessary and sufficient condition for the existence of an extension of a morphism from a submodule C ₀ of a partially ordered totally convex module C (with the ordered unit ball of the reals as codomain) to C, are proved. Moreover, the categories of partially ordered positively convex and superconvex modules are introduced and for both categories the Hahn–Banach Theorem is proved.     

On Translational Clouds for a Convex Body

For a d-dimensional convex body K let C(K) denote the minimum size of translational clouds for K. That is, C(K) is the minimum number of mutually non-overlapping translates of K which do not overlap K and block all the light rays emanating from any point of K. In this paper we prove the general upper bound . Furthermore, for an arbitrary centrally symmetric d-dimensional convex body S we show . Finally, for the d-dimensional ball B^d we obtain the bounds .     

Computational Experience with a New Class of Convex Underestimators: Box-constrained NLP Problems

In Akrotirianakis and Floudas (2004) we presented the theoretical foundations of a new class of convex underestimators for C ² nonconvex functions. In this paper, we present computational experience with those underestimators incorporated within a Branch-and-Bound algorithm for box-conatrained problems. The algorithm can be used to solve global optimization problems that involve C ² functions. We discuss several ways of incorporating the convex underestimators within a Branch-and-Bound framework. The resulting Branch-and-Bound algorithm is then used to solve a number of difficult box-constrained global optimization problems. A hybrid algorithm is also introduced, which incorporates a stochastic algorithm, the Random-Linkage method, for the solution of the nonconvex underestimating subproblems, arising within a Branch-and-Bound framework. The resulting algorithm also solves efficiently the same set of test problems.     

Convex hull theorem for multiply connected domains in the plane with an estimate of the quasiconformal constant

For any multiply connected domain Ω in R2, let S be the boundary of the convex hull in H3 of R2\Ω which faces Ω. Suppose in addition that there exists a lower bound l > 0 of the hyperbolic lengths of closed geodesics in Ω. Then there is always a K-quasiconformal mapping from S to Ω, which extends continuously to the identity on S = Ω, where K depends only on l. We also give a numerical estimate of K by using the parameter l.     

Generalized Convex Disjunctive Programming: Nonlinear Convex Hull Relaxation

Generalized Disjunctive Programming (GDP) has been introduced recently as an alternative to mixed-integer programming for representing discrete/continuous optimization problems. The basic idea of GDP consists of representing these problems in terms of sets of disjunctions in the continuous space, and logic propositions in terms of Boolean variables. In this paper we consider GDP problems involving convex nonlinear inequalities in the disjunctions. Based on the work by Stubbs and Mehrotra [21] and Ceria and Soares [6], we propose a convex nonlinear relaxation of the nonlinear convex GDP problem that relies on the convex hull of each of the disjunctions that is obtained by variable disaggregation and reformulation of the inequalities. The proposed nonlinear relaxation is used to formulate the GDP problem as a Mixed-Integer Nonlinear Programming (MINLP) problem that is shown to be tighter than the conventional big-M formulation. A disjunctive branch and bound method is also presented, and numerical results are given for a set of test problems.     

An Analog of the Krein--Mil'man Theorem for Strongly Convex Hulls in Hilbert Space

We prove the following theorem: in Hilbert space a closed bounded set is contained in the strongly convex R-hull of its R-strong extreme points. R-strong extreme points are a subset of the set of extreme points (it may happen that these two sets do not coincide); the strongly convex R-hull of a set contains the closure of the convex hull of the set.     

Convex minimization under Lipschitz constraints

We consider the problem of minimizing a convex functionf(x) under Lipschitz constraintsf _i (x)0,i=1,...,m. By transforming a system of Lipschitz constraintsf _i (x)0,i=l,...,m, into a single constraints of the formh(x)-x²0, withh(·) being a closed convex function, we convert the problem into a convex program with an additional reverse convex constraint. Under a regularity assumption, we apply Tuy's method for convex programs with an additional reverse convex constraint to solve the converted problem. By this way, we construct an algorithm which reduces the problem to a sequence of subproblems of minimizing a concave, quadratic, separable function over a polytope. Finally, we show how the algorithm can be used for the decomposition of Lipschitz optimization problems involving relatively few nonconvex variables.     

Convex programs with an additional reverse convex constraint

A method is presented for solving a class of global optimization problems of the form (P): minimizef(x), subject toxD,g(x)0, whereD is a closed convex subset ofR ⁿ andf,g are convex finite functionsR ⁿ . Under suitable stability hypotheses, it is shown that a feasible point is optimal if and only if 0=max{g(x):xD,f(x)f( )}. On the basis of this optimality criterion, the problem is reduced to a sequence of subproblemsQ _k ,k=1, 2, ..., each of which consists in maximizing the convex functiong(x) over some polyhedronS _k . The method is similar to the outer approximation method for maximizing a convex function over a compact convex set.     

Convex Hulls of Integral Points

The convex hull of all integral points contained in a compact polyhedron C is obviously a compact polyhedron. If C is not compact, then the convex hull K of its integral points need not be a closed set. However, under some natural assumptions, K is a closed set and a generalized polyhedron. Bibliography: 11 titles.     

Computing minimum-area rectilinear convex hull and L-shape

We study the problems of computing two non-convex enclosing shapes with the minimum area; the L-shape and the rectilinear convex hull. Given a set of n points in the plane, we find an L-shape enclosing the points or a rectilinear convex hull of the point set with minimum area over all orientations. We show that the minimum enclosing shapes for fixed orientations change combinatorially at most O(n) times while rotating the coordinate system. Based on this, we propose efficient algorithms that compute both shapes with the minimum area over all orientations. The algorithms provide an efficient way of maintaining the set of extremal points, or the staircase, while rotating the coordinate system, and compute both minimum enclosing shapes in O(n²) time and O(n) space. We also show that the time complexity of maintaining the staircase can be improved if we use more space.     

Horospheres and Convex Bodies in n-Dimensional Hyperbolic Space

In n-dimensional Euclidean space, the measure of hyperplanes intersecting a convex domain is proportional to the (n–2)-mean curvature integral of its boundary. This question was considered by Santaló in hyperbolic space. In non-Euclidean geometry the totally geodesic hypersurfaces are not always the best analogue to linear hyperplanes. In some situations horospheres play the role of Euclidean hyperplanes.In dimensions n=2 and 3, Santaló proved that the measure of horospheres intersecting a convex domain is also proportional to the (n–2)-mean curvature integral of its boundary.In this paper we show that this analogy does not generalize to higher dimensions. We express the measure of horospheres intersecting a convex body as a linear combination of the mean curvature integrals of its boundary.     

Closed Characteristics on Asymmetric Convex Hypersurfaces in R2n and the Corresponding Pinching Conditions

Abstract In this paper, we construct first a new concrete example of asymmetric convex compact C ^1,1-hypersurfaces in R ²ⁿ possessing precisely n closed characteristics. Then we prove multiplicity results on the closed characteristics on convex compact hypersurfaces in R ²ⁿ pinched by not necessarily symmetric convex compact hypersurfaces. *Partially supported by the 973 Program of STM, Funds of EC of Jiangsu, the Natural Science Funds of Jiangsu (BK 2002023), the Post-doctorate Funds of China, and the NNSF of China (10251001) **Partially supported by the 973 Program of STM, NNSF, MCME, RFDP, PMC Key Lab of EM of China, S. S. Chern Foundation, and Nankai University     

Convex Hulls in Singular Spaces of Negative Curvature

The paper gives a simple example of a complete CAT(–1)-space containing a set S with the following property: the boundary at infinity CH(S)of the convex hull of S differs from S by an isolated point. In contrast to this it is shown that if S is a union of finitely many convex subsets of a complete CAT(–1)-space X, then CH(S) = S. Moreover, this identity holds without restrictions on S if CH is replaced by some notion of almost convex hull.     

Weakly Compact Composition Operators on Locally Convex Spaces

Let E be a complete, barrelled locally convex space, let V = (v_n) be an increasing sequence of strictly positive, radial, continuous, bounded weights on the unit disc 𝔻 of the complex plane, and let φ be an analytic self map on 𝔻. The composition operators C_φ : f → f ○ φ on the weighted space of holomorphic functions HV (𝔻, E) which map bounded sets into relatively weakly compact subsets are characterized. Our approach requires a study of wedge operators between spaces of continuous linear maps between locally convex spaces which extends results of Saksman and Tylli [31, 32], and a representation of the space HV (𝔻, E) as a space of operators which complements work by Bierstedt , Bonet and Galbis [4] and by Bierstedt and Holtmanns [6].     

Newton's Problem of the Body of Minimal Resistance in the Class of Convex Developable Functions

We investigate the minimization of Newton's functional for the problem of the body of minimal resistance with maximal height M > 0 [4] in the class of convex developable functions defined in a disc. This class is a natural candidate to find a (non–radial) minimizer in accordance with the results of [9]. We prove that the minimizer in this class has a minimal set in the form of a regular polygon with n sides centered in the disc, and numerical experiments indicate that the natural number n > 2 is a non–decreasing function of M. The corresponding functions all achieve a lower value of the functional than the optimal radially symmetric function with the same height M.     

Cardinalities of Primitive Fixing Systems for Convex Bodies

We establish the exact upper bound of cardinalities of primitive fixing systems for any compact, convex body M , depending on dim M and md, M . In a sense, this is the last touch to the theory of fixing systems for compact, convex bodies. Received October 14, 1998, and in revised form May 5, 1999.     

Counting the onion

Iteratively computing and discarding a set of convex hulls creates a structure known as an “onion.” In this paper, we show that the expected number of layers of a convex hull onion for n uniformly and independently distributed points in a disk is Θ(n^2/3). Additionally, we show that in general the bound is Θ(n^2/(d+1)) for points distributed in a d‐dimensional ball. Further, we show that this bound holds more generally for any fixed, bounded, full‐dimensional shape with a nonempty interior. © 2004 Wiley Periodicals, Inc. Random Struct. Alg., 2004     

Maximality properties for isometric interpolating sequences and sequences of trivial points in the spectrum of H∞

Let (x_n) be an isometric interpolating sequence or a sequence of trivial points in the spectrum of H^∞. It is shown that either every cluster point of that sequence has a maximal support set or there exists y ∈ M(H^∞+C) such that the support of x_n is contained in the support of y for infinitely many n. Similar results for Gleason parts are obtained, too. We also investigate the H^∞‐convex hulls of countable unions of support sets and show that whenever supp x ? supp y and x /∈ , then the H^∞‐convex hull of supp x does not meet . (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)