Abstract. We give an extension of Hua’s inequality in pre-Hilbert C∗-modules without using convexity or the classical Hua’s inequality. As a consequence, some known and new generalizations of this inequality are deduced. Providing a Jensen inequality in the content of Hilbert C∗-modules, another extension of Hua’s inequality is obtained. We also present an operator Hua’s inequality, which is eq...

We study the relations between boundaries for algebras of holomorphic functions on Banach spaces and complex convexity of their balls. In addition, we show that the Shilov boundary for algebras of holomorphic functions on an order continuous sequence space X is the unit sphere SX if X is locally c-convex. In particular, it is shown that the unit sphere of the OrliczLorentz sequence space λφ,w i...

Stationary interpolatory subdivision schemes which preserve shape properties such as convexity or monotonicity are constructed. The schemes are rational in the data and generate limit functions that are at least C 2. The emphasis is on a class of six-point convexity preserving subdivision schemes that generate C 2 limit functions. In addition, a class of six-point monotonicity preserving scheme...

Abstract. Let f ∈ C[−1, 1] change its convexity finitely many times, in the interval. We are interested in estimating the degree of approximation of f by polynomials which are coconvex with it, namely, polynomials that change their convexity exactly at the points where f does. We discuss some Jackson type estimates where the constants involved depend on the location of the points of change of c...

The study of nontraditional notions of visibility and convexity is a fruitful branch of geometry, which has found applications in vlsi design, graphics, and motion planning. Researchers have investigated multiple variations of these notions, including orthogonal, finitely oriented, and restricted-orientation visibility and convexity, as well as nesw and link convexity. Rawlins introduced the no...

We investigate the concepts of linear convexity and C-convexity in complex Banach spaces. The main result is that any C-convex domain is necessarily linearly convex. This is a complex version of the Hahn-Banach theorem, since it means the following: given a C-convex domain Ω in the Banach space X and a point p / ∈Ω, there is a complex hyperplane through p that does not intersect Ω. We also prov...

We investigate the concepts of linear convexity and C-convexity in complex Banach spaces. The main result is that any C-convex domain is necessarily linearly convex. This is a complex version of the Hahn-Banach theorem, since it means the following: given a C-convex domain Ω in the Banach space X and a point p / ∈Ω, there is a complex hyperplane through p that does not intersect Ω. We also prov...

In [4] Bollobás considers “The spread of an infection on a square grid”. Initially some squares of a square grid are infected. The infection spreads to an uninfected square if at least two of the neighbouring squares are infected. A natural combinatorial problem — solved using an elegant perimeter argument in [4] — is to determine the minimum number of initially infected squares such that event...

We prove that a closed, geodesically convex subset C of P 2 (R) is closed with respect to weak convergence in P 2 (R). This means that if (μn) ⊂ C is such that μn ⇀ μ in duality with continuous bounded functions and supn ∫ |x|dμn <∞, then μ ∈ C as well.

We extend a result of Minsky to show that for a map of a surface to a hyperbolic 3-manifold which is not necessarily π1-injective but is 2-incompressible rel a geodesic link with a definite tube radius, the set of noncontractible simple loops with a bounded length representatives is quasi-convex in the complex of curves.