Given n points in the plane, we study three optimization problems of computing an empty pseudo-triangle: we consider minimizing the perimeter, maximizing the area, and minimizing the longest maximal concave chain. We consider two versions of the problem: First, we assume that the three convex vertices of the pseudotriangle are given. Let n denote the number of points that lie inside the convex ...

Given a set P of n points in the plane, we wish to find a set Q ⊂ P of k points for which the convex hull conv(Q) has the minimum area. We solve this, and the related problem of finding a minimum area convex k-gon, in time O(n log n) and space O(n log n) for fixed k, almost matching known bounds for the minimum area triangle problem. Our algorithm is based on finding a certain number of nearest...

Note: The legends (Fig. 1) used here and in the accompanying video are sligtly different from figures shown in the main paper. Motion segmentation results are shown by shade in correponding color of the convex-hull formed of the feature points segmented as independently moving. Reconstructed 3D static world points are colored depending on height from the estimated ground-plane. Trajectories and...

We provide an efficient data structure for dynamically maintaining a ham-sandwich cut of two (possibly overlapping) point sets in the plane, with a bounded number of convex-hull peeling layers. 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(c log n) tim...

In this article we discuss lattice convexity and concavity of Calderón-Lozanovskii space Eφ , generated by a quasi-Banach space E and an increasing Orlicz function φ. We give estimations of convexity and concavity indices of Eφ in terms of Matuszewska-Orlicz indices of φ as well as convexity and concavity indices of E. In the case when Eφ is a rearrangement invariant space we also provide some ...

Let p be a positive number. Consider the probability measure γ p with density ϕ p (y) = c n,p e − |y| p p. We show that the maximal surface area of a convex body in R n with respect to γ p is asymptotically equivalent to C (p)n 3 4 − 1 p , where the constant C (p) depends on p only. This is a generalization of results due to Ball (1993) [1] and Nazarov (2003) [9] in the case of the standard Gau...

We first obtain some properties of a fundamentally nonexpansive self-mapping on a nonempty subset of a Banach space and next show that if the Banach space is having the Opial condition, then the fixed points set of such a mapping with the convex range is nonempty. In particular, we establish that if the Banach space is uniformly convex, and the range of such a mapping is bounded, closed and con...

The affine surface area was introduced by Blaschke [B] for convex bodies in R with sufficently smooth boundary and by Leichtweiss [L1] for convex bodies in R with sufficently smooth boundary. As it occurs naturally in many important questions so for example in the approximation of convex bodies by polytopes ( see the survey article of Gruber [G] ) or in a priori estimates for PDEs [Lu2] one wan...

Given a planar convex set C, we give sublinear approximation algorithms to determine approximations of the largest axially symmetric convex set S contained in C, and the smallest such set S′ that contains C. More precisely, for any ε > 0, we find an axially symmetric convex polygon Q ⊂ C with area |Q| > (1 − ε)|S| and we find an axially symmetric convex polygon Q′ containing C with area |Q′|< (...

Many data sets are visualized effectively with area proportional Venn diagrams, where the area of the regions is in proportion to a defined specification. In particular, Venn diagrams with three intersecting curves are considered useful for visualizing data in many applications, including bioscience, ecology and medicine. To ease the understanding of such diagrams, using restricted ‘nice’ shape...