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 ...

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 ...

Color gamut characterization is an essential step in the ICC profile generation for imaging devices. It is also needed as part of evaluation and comparison of image quality achievable by different imaging systems. A novel technique is presented for characterizing the color gamut of an imaging device via ray tracing in the CIELAB color space. Initially, the convex hull of the set of measurement ...

There exists a surface of a convex polyhedron P and a partition L of P into geodesic convex polygons such that there are no connected"edge"unfoldings of P without self-intersections (whose spanning tree is a subset of the edge skeleton of L).

If F is a vector space over an ordered division ring, C a convex subset of V and L the lattice of convex subsets of C, then we call L a convexity lattice. We give necessary and sufficient conditions for an abstract lattice to be a convexity lattice in the finite dimensional case.

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.