Representation of nonnegative convex polynomials

We provide a specific representation of convex polynomials nonnegative on a convex (not necessarily compact) basic closed semi-algebraic set . Namely, they belong to a specific subset of the quadratic module generated by the concave polynomials that define . Received: 15 December 2007     

An Internal Polya Inequality for ℂ-Convex Domains in ℂn

Mathematical Notes - Let K ? ? be a polynomially convex compact set, f be a function analytic in a domain $$\overline{\mathbb{C}} \backslash K$$ with Taylor expansion $$f(z) =...     

Convex sets with semidefinite representation

We provide a sufficient condition on a class of compact basic semialgebraic sets for their convex hull co(K) to have a semidefinite representation (SDr). This SDr is explicitly expressed in terms of the polynomials g _j that define K. Examples are provided. We also provide an approximate SDr; that is, for every fixed , there is a convex set such that (where B is the unit ball of ), and has an explicit SDr in terms of the g _j ’s. For convex and compact basic semi-algebraic sets K defined by concave polynomials, we provide a simpler explicit SDr when the nonnegative Lagrangian L _f associated with K and any linear is a sum of squares. We also provide an approximate SDr specific to the convex case.      

The optimal statistical median of a convex set of arrays

We consider the following problem. A set of vectors is given. We want to find the convex combination such that the statistical median of z is maximum. In the application that we have in mind, are the historical return arrays of asset j and are the portfolio weights. Maximizing the median on a convex set of arrays is a continuous non-differentiable, non-concave optimization problem and it can be shown that the problem belongs to the APX-hard difficulty class. As a consequence, we are sure that no polynomial time algorithm can ever solve the model, unless P = NP. We propose an implicit enumeration algorithm, in which bounds on the objective function are calculated using continuous geometric properties of the median. Computational results are reported.     

On Well-posed Mutually Nearest and Mutually Furthest Point Problems in Banach Spaces

Let G be a non-empty closed(resp．bounded closed)boundedly relatively weakly compact subset in a strictly convex Kadec Banach space X．Let K(X)denote the space of all non-empty compact convex subsets of X endowed with the Hausdorff distance．Moreover,let KG(X)denote the closure of the set {A∈K(x)：A∩G=0}．We prove that the set of all A∈KG(X)(resp．A∈K(X)),such that the minimization (resp．maximization)problem min(A,G)(resp．max(A,G))is well posed,contains a dense Gδ-subset of KG(X)(resp．K(X))．thus extending the recent results due to Blasi,Myjak and Papini and Li．     

The minimum principle for affine functions and isomorphisms of continuous affine function spaces

Let X be a compact convex set and let ext X stand for the set of extreme points of X. We show that if $$f:X\rightarrow {\mathbb {R}}$$ is an affine function with the point of continuity property such that $$f\le 0$$ on $${\text {ext}}\,X$$, then $$f\le 0$$ on X. As a corollary of this minimum principle, we obtain a generalization of a theorem by C.H. Chu and H.B. Cohen by proving the following result. Let X, Y be compact convex sets such that every extreme point of X and Y is a weak peak point and let $$T:\mathfrak {A}^c(X)\rightarrow \mathfrak {A}^c(Y)$$ be an isomorphism such that $$\left\| T\right\| \cdot \left\| T^{-1}\right\| <2$$. Then $${\text {ext}}\,X$$ is homeomorphic to $${\text {ext}}\,Y$$.     

On Generic Well-posedness of Restricted Chebyshev Center Problems in Banach Spaces

Let B （resp. K, BC,KC） denote the set of all nonempty bounded （resp. compact, bounded convex, compact convex） closed subsets of the Banach space X, endowed with the Hausdorff metric, and let G be a nonempty relatively weakly compact closed subset of X. Let B° stand for the set of all F ∈B such that the problem （F, G） is well-posed. We proved that, if X is strictly convex and Kadec, the set KC ∩ B° is a dense Gδ-subset of KC ／ G. Furthermore, if X is a uniformly convex Banach space, we will prove more, namely that the set B ／B° （resp. K ／ B°, BC ／B°, KC ／ B°） is a-porous in B （resp. K,BC, KC）. Moreover, we prove that for most （in the sense of the Baire category） closed bounded subsets G of X, the set K ／ B° is dense and uncountable in K.     

Reconstruction of a plane convex body from the curvature of its boundary

Let denote the class of plane convex bodies having a width functionw, wherew′ is absolutely continuous. It is proved that a body in is determined (up to translation) by the radius of curvature function of its boundary. This result is then used for a characterization of the extreme (indecomposable) bodies in and for a density theorem for Reuleaux polygons in . The content of this paper is a revised version of a part of the Master of Science thesis written by the author under the supervision of Professor Micha A. Perles at the Hebrew University of Jerusalem and submitted in October, 1971.     

On a theorem of Helly

We consider a group of problems related to the well-known Helly theorem on the intersections of convex bodies. We introduce convex subsetsK(?) of a compact convex setK defined by the relation $$K(f) = co\left\{ {\frac{N}{{N + 1}}x + \frac{N}{{N + 1}}f(x)} \right\}{\text{ }}(x \in K \subset \mathbb{R}^N ),$$ where?: K→K are continuous mappings, and prove that the intersection ∩ _?∈F K(?) is not empty; hereF is the set of all continuous mappings?: K→K.     

对偶均值积分的Marcus-Lopes不等式

Milman曾提出过一个问题;在混合体积理论,是否存在Marcus-Lopes型和Bergstrom型不等式?即对R~n上任意凸体K与L且i=0,…,n-1,是否成立(W_i(K+L))/(W_i+1(K+L))≥(W_i(K))/(W_i+1(K))+(W_i(L))/(W_i+1(L))?这里W_i表示凸体的i次均值积分.当且仅当i=n-1或i=n-2时,这个问题是正确的,已被证明.作者考虑了一个对偶问题,证明了:若K与L是R~n上的星体,n-2≤i≤n-1且i∈R,则(W_i(K+L))/(W_i+1(K+L))≤(W_i(K))/(W_i+1(K))+(W_i(L))/(W_i+1(L))/(W_i+1(L))其中W_i表示星体的i次对偶均值积分.     

Shortest periodic billiard trajectories in convex bodies

We show that the length of any periodic billiard trajectory in any convex body is always at least 4 times the inradius of K; the equality holds precisely when the width of K is twice its inradius, e.g., K is centrally symmetric, in which case we prove that the shortest periodic trajectories are all bouncing ball (2-link) orbits.     

Complexity bounds for primal-dual methods minimizing the model of objective function

Mathematical Programming - We provide Frank–Wolfe ( $$\equiv $$ Conditional Gradients) method with a convergence analysis allowing to approach a primal-dual solution of convex optimization...     

Distance sets corresponding to convex bodies

Suppose that is a 0-symmetric convex body which denes the usual norm

Packing and Covering with Centrally Symmetric Convex Disks

Given a convex disk K (a convex compact planar set with nonempty interior), let δ _L (K) and θ _L (K) denote the lattice packing density and the lattice covering density of K, respectively. We prove that for every centrally-symmetric convex disk K we have that $$ 1\le\delta_L(K)\theta_L(K)\le1.17225\ldots $$ The left inequality is tight and it improves a 10-year old result.     

On convex triangle functions

We prove that the strongest (largest convex) solution of the functional inequality $$\tau \left( {\frac{{F + G}}{2},\frac{{H + K}}{2}} \right) \le \frac{{\tau (F,H) + \tau (G,K)}}{2},$$ whereF, G, H andK are arbitrary distribution functions, is the triangle function τ(F, G)(x) = Max(F(x) +G(x) ? 1, 0).     

Fejér and Hermite–Hadamard type results for H-invex functions with applications

Positivity - We investigate the class of H-invex functions including, e.g., the subclasses of convex, c-strongly convex, $$ \varphi $$ -uniformly convex and superquadratic functions. For H-invex...     

A $$\tau $$ τ -Conjecture for Newton Polygons

Foundations of Computational Mathematics - One can associate to any bivariate polynomial $$P(X,Y)$$ its Newton polygon. This is the convex hull of the points $$(i,j)$$ such that the monomial $$X^i...     

Parallelepipeds circumscribed about a convex body in 3-space

It is proved that each convex body K ⊂ ℝ³ of volume V(K) is contained in a parallelepiped of volume . Bibliography: 4 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 329, 2005, pp. 79–87.     

Trajectory of an Observer Tracking the Motion of an Object around a Convex Set in $${{\mathbb{R}}^{3}}$$

Doklady Mathematics - An object t moving in $${{\mathbb{R}}^{3}}$$ goes around a solid convex set along the shortest path $$\mathcal{T}$$ under observation. The task of an observer f (moving at the...