Let ∆ be a thick dual polar space and F a convex subspace of diameter at least 2 of ∆. Every hyperplane G of the subgeometry F̃ of ∆ induced on F will give rise to a hyperplane H of ∆, the so-called extension of G. We show that F and G are in some sense uniquely determined by H. We also consider the following problem: if e is a full projective embedding of ∆ and if eF is the full embedding of F̃ ...

In this paper, we present various functional means in the sense of convex analysis. In particular, a logarithmic mean involving convex functionals, extending the scalar one, is introduced. In the quadratic case, our functional approach implies immediately that of positive operators. Some examples, illustrating theoretical results and showing the interest of our functional approach, are discussed.

We consider the class of semi-stable positive solutions to semilinear equations −∆u = f(u) in a bounded domain Ω ⊂ Rn of double revolution, that is, a domain invariant under rotations of the first m variables and of the last n−m variables. We assume 2 ≤ m ≤ n− 2. When the domain is convex, we establish a priori Lp and H1 0 bounds for each dimension n, with p = ∞ when n ≤ 7. These estimates lead...

Separable sublinear functions are used to provide upper bounds on the recourse function of a stochastic program. The resulting problem's objective involves the inf-convolution of convex functions. A dual of this problem is formulated to obtain an implementable procedure to calculate the bound. Function evaluations for the resulting convex program only require a small number of single integratio...

The concept of neighbor system, introduced by Hartvigsen (2009), is a set of integral vectors satisfying a certain combinatorial property. In this paper, we reveal the relationship of neighbor systems with jump systems and with bisubmodular polyhedra. We firstly prove that for every neighbor system, there exists a jump system which has the same neighborhood structure as the original neighbor sy...

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, ...

For functions defined on integer lattice points, discrete versions of the Hessian matrix have been considered in various contexts. In discrete convex analysis, for example, certain combinatorial properties of the discrete Hessian matrices are known to characterize M-convex and L-convex functions, which can be extended to convex functions in real variables. The relationship between convex extens...

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...

We consider a class of integer-valued discrete convex functions, called BS-convex functions, defined on integer lattices whose affinity domains are sets of integral points of integral bisubmodular polyhedra. We examine discrete structures of BSconvex functions and give a characterization of BS-convex functions in terms of their convex conjugate functions by means of (discordant) Freudenthal sim...

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...