Given a compact subset F of R2, the visible part VθF of F from direction θ is the set of x in F such that the half-line from x in direction θ intersects F only at x. It is suggested that if dimH F ≥ 1, then dimH VθF = 1 for almost all θ, where dimH denotes Hausdorff dimension. We confirm this when F is a self-similar set satisfying the convex open set condition and such that the orthogonal proj...

This paper provides new versions of the Farkas lemma characterizing those inequalities of the form f(x) ≥ 0 which are consequences of a composite convex inequality (S ◦ g)(x) ≤ 0 on a closed convex subset of a given locally convex topological vector space X, where f is a proper lower semicontinuous convex function defined on X, S is an extended sublinear function, and g is a vector-valued S-con...

Eszter Klein’s theorem claims that among any 5 points in the plane, no three collinear, there is the vertex set of a convex quadrilateral. An application of Ramsey’s theorem then yields the classical Erdős–Szekeres theorem [19]: For every integer n ≥ 3 there is an N0 such that, among any set of N ≥ N0 points in general position in the plane, there is the vertex set of a convex n-gon. Let f(n) d...

In stochastic convex optimization the goal is to minimize a convex function F (x) . = Ef∼D[f(x)] over a convex set K ⊂ R where D is some unknown distribution and each f(·) in the support of D is convex over K. The optimization is commonly based on i.i.d. samples f, f, . . . , f from D. A standard approach to such problems is empirical risk minimization (ERM) that optimizes FS(x) . = 1 n ∑ i≤n f...

and Applied Analysis 3 Definition 2.1. Let F : K → 2Rn be a set-valued mapping. F is said to be i monotone on K if for each pair of points x, y ∈ K and for all x∗ ∈ F x and y∗ ∈ F y , 〈y∗ − x∗, y − x〉 ≥ 0, ii maximal monotone on K if, for any u ∈ K, 〈ξ − x∗, u − x〉 ≥ 0 for all x ∈ K and all x∗ ∈ F x implies ξ ∈ F u , iii quasimonotone on K if for each pair of points x, y ∈ K and for all x∗ ∈ F ...

and Applied Analysis 3 xn ⇀ x ∈ X and ‖xn‖ → ‖x‖, then xn → x. It is known that if X is uniformly convex, then X has the Kadec-Klee property. The normalized duality mapping J from X to X∗ is defined by Jx { x∗ ∈ X∗ : 〈x, x∗〉 ‖x‖ ‖x∗‖2 } 2.3 for any x ∈ X. We list some properties of mapping J as follows. i If X is a smooth Banach space with Gâteaux differential norm , then J is singlevalued and ...

We study the determination of finite subsets of the integer lattice Zn, n ≥ 2, by X-rays. In this context, an X-ray of a set in a direction u gives the number of points in the set on each line parallel to u. For practical reasons, only X-rays in lattice directions, that is, directions parallel to a nonzero vector in the lattice, are permitted. By combining methods from algebraic number theory a...

A set is called semidefinite representable or semidefinite programming (SDP) representable if it can be represented as the projection of a higher dimensional set which is represented by some Linear Matrix Inequality (LMI). This paper discuss the semidefinite representability conditions for convex sets of the form SD(f) = {x ∈ D : f(x) ≥ 0}. Here D = {x ∈ R n : g1(x) ≥ 0, · · · , gm(x) ≥ 0} is a...

• F (x) is a convex quadratic, i.e. F (x) = xTMx + vTx (with M 0 and v ∈ Rn). • P ⊆ Rn is a convex set over which we can efficiently optimize F , • K ⊆ Rn is a non-convex set with “special structure”. • Typically, n could be quite large. A standard approach to solving this problem would start by solving a convex relaxation to F , thereby obtaining a lower bound on F z. However, when K is comple...

We show that there is a convex ring R = Ω \Ω ⊂ R in which there exists a solution u to a semilinear partial differential equation ∆u = f(u), u = −1 on ∂Ω, u = 1 on ∂Ω, with level sets, not all convex. Moreover every bounded solution u has at least one nonconvex level set. In our construction, the nonlinearity f , is non-positive, and smooth. AMS Classification: 35J60, 35R35.