1. Introduction 2. Order and type in classical complex analysis 3. Relative order and type of convex functions 4. Order and type in duality 5. The infimal convolution 6. The order of an entire function 7. A geometric characterisation of the relative order 8. An extension problem for holomorphic functions 9. Whittaker's decomposition theorem 10. Inequalities for the infimal convolution 11. Inequ...

An empty pentagon in a point set P in the plane is a set of five points in P in strictly convex position with no other point of P in their convex hull. We prove that every finite set of at least 328` points in the plane contains an empty pentagon or ` collinear points. This is optimal up to a constant factor since the (` − 1) × (` − 1) grid contains no empty pentagon and no ` collinear points. ...

We consider the problem of minimizing a convex function plus a polynomial p over a convex body K. We give an algorithm that outputs a solution x whose value is within rangeK(p) of the optimum value, where rangeK(p) = supx∈K p(x)− infx∈K p(x). When p depends only on a constant number of variables, the algorithm runs in time polynomial in 1/ , the degree of p, the time to round K and the time to ...

Let C be a closed convex subset of a Banach space E. A mapping T on C is called a nonexpansive mapping if ‖Tx−Ty‖ ≤ ‖x− y‖ for all x, y ∈ C. We denote by F(T) the set of fixed points of T . Kirk [17] proved that F(T) is nonempty in the case that C is weakly compact and has normal structure. See also [2, 3, 5, 6, 11] and others. Convergence theorems to fixed points are also proved by many author...

We deal with different properties of a smooth and strictly convex body that depend on the behavior of the planar sections of the body parallel to and close to a given tangent plane. The first topic is boundary points where any given convex domain in the tangent plane can be approximated by a sequence of suitably rescaled planar sections (so-called p-universal points). In the second topic, the g...

In this paper, we give twoweak conditions for a lower semi-continuous function on the n-dimensional Euclidean space Rn to be a convex function. We also present some results for convex functions, strictly convex functions, and quasi-convex functions.

We consider the following basic geometric problem: Given ∈ (0, 1/2), a 2-dimensional figure that consists of a black object and a white background is -far from convex if it differs in at least an fraction of the area from every figure where the black object is convex. How many uniform and independent samples from a figure that is -far from convex are needed to detect a violation of convexity wi...

where f(.) : Γf → R is an unknown, not necessarily smooth, multivariate and λ-strongly convex function, with Γf its convex definition domain. The algorithm is not allowed to accurately sample f(.) by any means since f(.) itself is unknown. Instead the algorithm can call stochastic oracles ω̃(.) at chosen points x̃1, . . . , x̃n, which are unbiased and independent probabilistic estimators of the fi...

In a partially ordered semigroup with the duality (or polarity) transform, it is possible to define a generalisation of continued fractions. General sufficient conditions for convergence of continued fractions are provided. Two particular applications concern the cases of convex sets with the Minkowski addition and the polarity transform and the family of non-negative convex functions with the ...

The present paper gives characterizations of radially u.s.c. convex and pseudoconvex functions f : X → R defined on a convex subset X of a real linear space E in terms of first and second-order upper Dini-directional derivatives. Observing that the property f radially u.s.c. does not require a topological structure of E, we draw the possibility to state our results for arbitrary real linear spa...