We consider a variant of Heilbronn’s triangle problem by asking for fixed integers d, k ≥ 2 and any integer n ≥ k for a distribution of n points in the d-dimensional unit cube [0, 1] such that the minimum volume of the convex hull of k points among these n points is as large as possible. We show that there exists a configuration of n points in [0, 1], such that, simultaneously for j = 2, . . . ...

A path P = v1, . . . , vt is a triangle path (respectively, monophonic path) of G if no edges exist joining vertices vi and vj of P such that |j − i| > 2; (respectively, |j − i| > 1). A set of vertices S is convex in the triangle path convexity (respectively, monophonic convexity) of G if the vertices of every triangle path (respectively, monophonic path) joining two vertices of S are in S. The...

Let B be the set of n-dimensional binary vectors and let V be a subset of m of its elements. We give an extended formulation of the convex hull of B \ V which is polynomial in n and m. In developing this result, we give a two-sided extension of the result in [3] for knapsack sets with superincreasing coefficients.

A coarse outer hull of a mesh is a good tool used in computer graphics to reduce algorithm complexity, especially, in applications such as collision detection or ray-tracing. It is often required that the hull has some very specific parameters concerning its shape and quality since these influence general flexibility and numerical stability of the algorithms. This paper overviews existing probl...

A necessary and sufficient condition on the support of a generic unmixed bivariate polynomial system is identified such that for polynomial systems with such support, the Dixon resultant formulation produces their resultants. It is shown that Sylvester-type matrices, called Dixon dialytic matrices, can also be obtained for such polynomial systems. These results are shown to be a generalization ...

We present two algorithms that compute the Newton polytope of a polynomial f defining a hypersurface H in Cn using numerical computation. The first algorithm assumes that we may only compute values of f—this may occur if f is given as a straightline program, as a determinant, or as an oracle. The second algorithm assumes that H is represented numerically via a witness set. That is, it computes ...

Matrix functions are used in many areas of linear algebra and arise in numerical applications in science and engineering. In this paper, we introduce an effective approach for determining matrix function f(A)=g(q(A)) of a square matrix A, where q is a polynomial function from a degree of m and also function g can be a transcendental function. Computing a matrix function f(A) will be time- consu...