In this paper, we give an algorithm for output-sensitive construction of an f-face convex hull of a set of n points in general position in E 4. Our algorithm runs in O((n + f)log 2 f) time and uses O(n + f) space. This is the rst algorithm within a polylogarithmic factor of optimal O(n log f + f) time over the whole range of f. By a standard lifting map, we obtain output-sensitive algorithms fo...

We study the weak* lower semicontinuity properties of functionals of the form F (u) = ess sup x∈Ω f(x,Du(x)) where Ω is a bounded open set of R and u ∈ W 1,∞(Ω). Without a continuity assumption on f(·, ξ) we show that the supremal functional F is weakly* lower semicontinuous if and only if it is a level convex functional (i.e. it has convex sub levels). In particular if F is weakly* lower semic...

Let F be a family of positive homothets (or translates) of a given convex body K in Rn. We investigate two approaches to measuring the complexity of F . First, we find an upper bound on the transversal number τ(F) of F in terms of n and the independence number ν(F). This question is motivated by a problem of Grünbaum [2]. Our bound τ(F) ≤ 2n (2n n ) (n logn+ log logn+ 5n)ν(F) is exponential in ...

where q~ is a c losed, convex func t ion def ined on R n, having values in ~ and S is a c losed, convex set in ~n. We write S for the op t ima l so lu t ion set o f (1), S : = arg minxes cb(x) and assume this set to be non-empty , in o rde r tha t a p ro jec t ion o p e r a t i o n on to this set is well def ined. In o rde r to s impl i fy our analysis , let us define , b ~ ( x ) := ¢~(x )+~, (...

If X is a convex surface in a Euclidean space, then the squared intrinsic distance function dist(x, y) is DC (d.c., delta-convex) on X×X in the only natural extrinsic sense. An analogous result holds for the squared distance function dist(x,F ) from a closed set F ⊂ X. Applications concerning r-boundaries (distance spheres) and ambiguous loci (exoskeletons) of closed subsets of a convex surface...

The notion of a generalized convex space we work with in this paper was introduced by Park and Kim in [10]. In generalized convex spaces, many results on fixed points, coincidence points, equilibrium problems, variational inequalities, continuous selections, saddle points, and others have been obtained, see, for example, [6, 8, 10–13]. In this paper, we obtain an almost coincidence point theore...

One of our results: let X be a finite set on the plane, 0 < g < 1, then there exists a set F (a weak g-net) of size at most 7/e 2 such that every convex set containing at least e\X\ elements of X intersects F. Note that the size of F is independent of the size of X.

Let Ω ⊂ R be a bounded open set. Given 2 ≤ m ≤ n, we construct a convex function φ : Ω → R whose gradient f = ∇φ is a Hölder continuous homeomorphism, f is the identity on ∂Ω, the derivative Df has rank m − 1 a.e. in Ω and Df is in the weak L space L. The proof is based on convex integration and staircase laminates.