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the aim of this paper is to prove some inequalities for p-valent meromorphic functions in thepunctured unit disk δ* and find important corollaries.

Given a convex polygon with n vertices in the plane, we are interested in triangulations of its interior, i.e., maximal sets of nonintersecting diagonals that subdivide the interior of the polygon into triangles. The MaxMin area triangulation is the triangulation of the polygon that maximizes the area of the smallest triangle in the triangulation. Similarly, the MinMax area triangulation is the...

The main purposes of this paper are to establish some new Brunn– Minkowski inequalities for width-integrals of mixed projection bodies and affine surface area of mixed bodies, together with their inverse forms.

Given a set P = {P0, . . . , Pk−1} of k convex polygons having n vertices in total in the plane, we consider the problem of finding k translations τ0, . . . , τk−1 of P0, . . . , Pk−1 such that the translated copies τiPi are pairwise disjoint and the area or the perimeter of the convex hull of ⋃k−1 i=0 τiPi is minimized. When k = 2, the problem can be solved in linear time but no previous work ...

Abstract. Let Ω ⊆ R have minimal Gaussian surface area among all sets satisfying Ω = −Ω with fixed Gaussian volume. Let A = Ax be the second fundamental form of ∂Ω at x, i.e. A is the matrix of first order partial derivatives of the unit normal vector at x ∈ ∂Ω. For any x = (x1, . . . , xn+1) ∈ R, let γn(x) = (2π)−n/2e 2 1+···+x 2 n+1. Let ‖A‖2 be the sum of the squares of the entries of A, and...

Let P be a set of n points in the plane and O be a set of k lines passing through the origin. We show: (1) How to compute the O-hull of P in Θ(n log n) time and O(n) space, (2) how to compute and maintain the rotated hull OHθ(P ) for θ ∈ [0, 2π) in O(kn log n) time and O(kn) space, and (3) how to compute in Θ(n log n) time and O(n) space a value of θ for which the rectilinear convex hull, RHθ(P...

Article history: Received 21 February 2008 Accepted 24 February 2009 Available online 24 March 2009 Communicated by T. Asano

We provide an efficient data structure for dynamically maintaining a ham-sandwich cut of two overlapping point sets in convex position in the plane. The ham-sandwich cut of S1 and S2 is a line that simultaneously bisects the area, perimeter or vertex count of both point sets. Our algorithm supports insertion and deletion of vertices in O(log n) time, and area, perimeter and vertex-count queries...

A digraph D is connected if the underlying undirected graph of D is connected. A subgraph H of an acyclic digraph D is convex if there is no directed path between vertices of H which contains an arc not in H. We find the minimum and maximum possible number of connected convex subgraphs in a connected acyclic digraph of order n. Connected convex subgraphs of connected acyclic digraphs are of int...