Let Ω be the upper half plane {(x, y) ∈ R, x > 0, y ∈ R}. Define the Laplacian on Ω to be ∆D = ∂ 2 x + (1 + x)∂ 2 y , together with Dirichlet boundary conditions on ∂Ω: one may easily see that Ω, with the metric inherited from ∆D, is a strictly convex domain. We shall prove that, in such a domain Ω, Strichartz estimates for the wave equation suffer losses when compared to the usual case Ω = R, ...

A restricted-orientation convex set, also called an O-convex set, is a set of points whose intersection with lines from some fixed set is empty or connected. The notion of O-convexity generalizes standard convexity and orthogonal convexity. We explore some of the basic properties of O-convex sets in two and higher dimensions. We also study O-connected sets, which are a subclass of O-convex sets...

Suppose that K is a nonempty closed convex subset of a real uniformly convex Banach space E , which is also a nonexpansive retract of E with nonexpansive retraction P . Let {Ti : i ∈ I } be N nonself asymptotically nonexpansive mappings from K to E such that F = {x ∈ K : Ti x = x, i ∈ I } 6= φ, where I = {1, 2, . . . , N }. From arbitrary x0 ∈ K , {xn} is defined by xn = P((1− αn)xn−1 + αnTn(PT...

Let Ω ⊆ Rn be a strictly convex domain and let φ ∈ C2(Ω) be a convex function such that λ ≤ detD2φ ≤ Λ in Ω. The linearized Monge– Ampère equation is LΦu = trace(ΦD u) = f, where Φ = (detD2φ)(D2φ)−1 is the matrix of cofactors of D2φ. We prove that there exist p > 0 and C > 0 depending only on n, λ,Λ, and dist(Ω′,Ω) such that ‖Du‖Lp(Ω′) ≤ C(‖u‖L∞(Ω) + ‖f‖Ln(Ω)) for all solutions u ∈ C2(Ω) to the...

We prove that among all doubly connected domains of Rn bounded by two spheres of given radii, Z(t), the trace of the heat kernel with Dirichlet boundary conditions, achieves its minimum when the spheres are concentric (i.e., for the spherical shell). The supremum is attained when the interior sphere is in contact with the outer sphere. This is shown to be a special case of a more general theore...

We present coincidence points results for multivalued f nonexpansive mappings in the setting of nonstarshaped domain of q-normed space which is not necessarily a locally convex space. As application, an invariant approximation result is also obtained. Our results improve and extend the results of Bano, Khan and Latif [1], Hussain [4], Latif and Tweddle [7], Rhoades [11], Sahab, Khan and Sessa [...

It is shown that every plane compact convex set /f with an interior point admits a covering of the plane with density smaller than or equal to 8(2\/3 — 3)/3 = 1.2376043 For comparison, the thinnest covering of the plane with congruent circles is of density 2n/\Z21 = 1.209199576... (see R. Kershner [3]), which shows that the covering density bound obtained here is close to the best possible. It ...

In 1950 Bang proposed a conjecture which became known as “the plank conjecture”: Suppose that a convex set S contained in the unit cube of <n and touching all its sides is covered by planks. (A plank is a set of the form {(x1, . . . , xn): xj ∈ I } for some j ∈ {1, . . . , n} and a measurable subset I of [0, 1]. Its width is defined as |I |.) Then the sum of the widths of the planks is at least...

Given a convex set and an interior point close to the boundary, we prove the existence of a supporting hyperplane whose distance to the point is controlled, in a dimensionally quantified way, by the thickness of the convex set in the orthogonal direction. This result has important applications in the regularity theory for Monge-Ampère type equations arising in optimal transportation.