This is a survey of algorithmic results in the theory of “discrete convex analysis” for integer-valued functions defined on integer lattice points. The theory parallels the ordinary convex analysis, covering discrete analogues of the fundamental concepts such as conjugacy, the Fenchel min-max duality, and separation theorems. The technical development is based on matroid-theoretic concepts, in ...

Let H(k, l) denote the smallest integer such that any set of H(k, l) points in the plane, no three on a line, contains an empty convex k-gon and an empty convex l-gon, which are disjoint, that is, their convex hulls do not intersect. Hosono and Urabe [JCDCG, LNCS 3742, 117-122, 2004] proved that 12 ≤ H(4, 5) ≤ 14. Very recently, using a Ramsey-type result for disjoint empty convex polygons prov...

This paper uses convex bodies to study line bundles in the setting of Arakelov theory. The treatment is parallel to [Yu2], but the content is independent. The method of constructing a convex body in Euclidean space, now called “Okounkov body”, from a given algebraic linear series was due to Okounkov [Ok1, Ok2], and was explored systematically by Kaveh–Khovanskii [KK] and Lazarsfeld–Mustaţǎ [LM]...

In this note we show that a non-degenerated polytope in IRn with n+k, 1 ≤ k < n, vertices is far from any symmetric body. We provide the asymptotically sharp estimates for the asymmetry constant of such polytopes. 0 Introduction and notations The canonical Euclidean inner product in IR is denoted by 〈·, ·〉, the norm in `p is denoted by ‖ · ‖p, 1 ≤ p ≤ ∞. By a convex body K ⊂ IR we shall always ...

conclusions in this study, the l-lps immune response of bovine cells was significant, indicating that tlr2 is the predominant receptor for l-lps. due to l-lps endotoxin-like activity, we found a strategy through using bmap-28 to prevent l-lps-induced tlr2-dependent immune activation in bovine cells. materials and methods in this study, l-lps-induced proinflammatory cytokine production in bovine...

This lecture describes a data structure for representing convex polytopes and a divide and conquer algorithm for computing convex hull in 3 dimensions. Let S be a set of n points in . Convex hull of S (CH(S)) is the smallest convex polytope that contains all n points. Since the boundary of this polytope is planar, it can be efficiently represented by the data structure described in the next sec...

A smooth curve γ : [0, 1] → S2 is locally convex if its geodesic curvature is positive at every point. J. A. Little showed that the space of all locally convex curves γ with γ(0) = γ(1) = e1 and γ (0) = γ(1) = e2 has three connected components L−1,c, L+1, L−1,n. The space L−1,c is known to be contractible. In this paper we prove that L+1 and L−1,n are homotopy equivalent to ΩS3 ∨ S2 ∨ S6 ∨ S10 ...