We show that the fundamental objects of the Lp-Brunn-Minkowski theory, namely the Lp-affine surface areas for a convex body, are closely related to information theory: they are exponentials of Rényi divergences of the cone measures of a convex body and its polar. We give geometric interpretations for all Rényi divergences Dα, not just for the previously treated special case of relative entropy ...

What is the minimal area A(n) a convex lattice polygon with n vertices can have? The first to answer this question was G.E. Andrews [1]. He proved that A(n)≥cn3 with some universal constant c. V.I. Arnol’d arrived to the same question from another direction [2], and proved the same estimate. Further proofs are due to W. Schmidt [10], Bárány–Pach [3]. The best lower bound comes form Rabinowitz [...

the rational cubic function with three parameters has been extended to rational bi-cubic function to visualize the shape of regular convex surface data. the rational bi-cubic function involves six parameters in each rectangular patch. data dependent constraints are derived on four of these parameters to visualize the shape of convex surface data while other two are free to refine the shape of s...

Abstract We show a decomposition that allows to compute the number of convex polyominoes area n and degree convexity at most 2 (the so-called Z-convex polyominoes) in polynomial time.

Let K ⊂ R be a smooth convex set and let Pλ be a Poisson point process on R of intensity λ. The convex hull of Pλ ∩ K is a random convex polytope Kλ. As λ → ∞, we show that the variance of the number of k-dimensional faces of Kλ, when properly scaled, converges to a scalar multiple of the affine surface area of K. Similar asymptotics hold for the variance of the number of k-dimensional faces fo...

Two families of general affine surface areas are introduced. Basic properties and affine isoperimetric inequalities for these new affine surface areas as well as for Lφ affine surface areas are established. 2000 AMS subject classification: Primary 52A20; Secondary 53A15. Finding the right notion of affine surface area was one of the first questions asked within affine differential geometry. At ...

in this note we first redefine the notion of a fuzzy hypervectorspace (see [1]) and then introduce some further concepts of fuzzy hypervectorspaces, such as fuzzy convex and balance fuzzy subsets in fuzzy hypervectorspaces over valued fields. finally, we briefly discuss on the convex (balanced)hull of a given fuzzy set of a hypervector space.

Convex optimization is a branch of mathematics dealing with nonlinear optimization problems with additional geometric structure. This area has been the focus of considerable recent research due to the fact that convex optimization problems are scalable and can be efficiently solved by interior-point methods. Over the last ten years or so, convex optimization has found new applications in many a...

Graphs are often used to represent relational information. As the amount of information that we want to visualize becomes larger and more complicated, classical graph model tends to be insuucient. In this paper, we introduce and show how to draw a practical and simple graph structure known as clustered graphs. We present an algorithm which produces planar, straight-line, convex drawings of clus...

knowing the fact that the main weakness of the most standard methods including k-means and hierarchical data clustering is their sensitivity to initialization and trapping to local minima, this paper proposes a modification of convex data clustering  in which there is no need to  be peculiar about how to select initial values. due to properly converting the task of optimization to an equivalent...