It is shown that every equi-affine invariant and upper semicontinuous valuation on the space of convex discs is a linear combination of the Euler characteristic, area, and affine length. Asymptotic formulae for approximation of convex discs by polygons are derived, extending results of L. Fejes Tóth from smooth convex discs to general convex discs. 1991 AMS subject classification: 52A10, 53A15,...

Parameterization of a 3D triangular mesh is a fundamental problem in various applications of meshes. The convex combination approach is widely used for parameterization because of its good properties, such as fast computation and one-to-one embedding. However, the approach has a drawback: most boundary triangles have high distortion in the embedding compared with interior ones. In this paper, w...

Given a collection of minimal graphs,M1,M2, . . . ,Mn, with isothermal parametrizations in terms of the Gauss map and height differential, we give sufficient conditions onM1,M2, . . . ,Mn so that a convex combination of themwill be a minimal graph. We will then provide two examples, taking a convex combination of Scherk’s doubly periodic surface with the catenoid and Enneper’s surface, respecti...

An inequality g{x) 2i 0 is often said to be a reverse convex constraint if the function g is continuous and convex. The feasible regions for linear program with an additional reverse convex constraint are generally non-convex and disconnected. In this paper a convergent algorithm for solving such a linear problem is proposed. The method is based upon a combination of the branch and bound proced...

We consider a hypersurface of dimension d imbedded in a d + 1 dimensional space. For each x 2 Z d , let t (x) 2 R be the height of the surface at site x at time t. At rate 1 the x-th height is updated to a random convex combination of the heights of thèneighbors' of x. The distribution of the convex combination is translation invariant and does not depend on the heights. This motion, named the ...

λ1 + . . .+ λm = 1, then we say that y is an affine combination of y1, . . . ,ym ∈Y . If, in addition, λi ≥ 0 for 1 ≤ i ≤ m, then we say that y is a convex combination of y1, . . . ,ym ∈ Y . A convex set is any subset of Rn that is closed under the operation of taking convex combinations. In fact, it can be shown that a subset X is convex if and only if for all x0,x1 ∈ X and 0 ≤ λ ≤ 1, the poin...

This paper shows the equivalence between Murota’s L-convex functions and Favati and Tardella’s submodular integrally convex functions: For a submodular integrally convex function g(p1, . . . , pn), the function g̃ defined by g̃(p0, p1, . . . , pn) = g(p1 − p0, . . . , pn − p0) is an L-convex function, and vice versa. This fact implies, in combination with known results for L-convex functions, tha...

Introduction. A classical result in the theory of convex polyhedra is that every bounded polyhedral convex set can be expressed either as the intersection of half-spaces or as a convex combination of extreme points. It is becoming increasingly apparent that a full understanding of a class of convex polyhedra requires the knowledge of both of these characterizations. Perhaps the earliest and nea...