scholarly journals Colorful Simplicial Depth, Minkowski Sums, and Generalized Gale Transforms

2017 ◽  
Vol 2019 (6) ◽  
pp. 1894-1919
Author(s):  
Karim A Adiprasito ◽  
Philip Brinkmann ◽  
Arnau Padrol ◽  
Pavel Paták ◽  
Zuzana Patáková ◽  
...  
Keyword(s):  
Convex Hull ◽  
Upper Bound ◽  
Finite Sets ◽  
Combinatorial Topology ◽  
Normal Surfaces ◽  
Simplicial Depth ◽  
Minkowski Sums

Abstract The colorful simplicial depth of a collection of $d+1$ finite sets of points in Euclidean $d$-space is the number of choices of a point from each set such that the origin is contained in their convex hull. We use methods from combinatorial topology to prove a tight upper bound on the colorful simplicial depth. This implies a conjecture of Deza et al. [7]. Furthermore, we introduce colorful Gale transforms as a bridge between colorful configurations and Minkowski sums. Our colorful upper bound then yields a tight upper bound on the number of totally mixed facets of certain Minkowski sums of simplices. This resolves a conjecture of Burton [6] in the theory of normal surfaces.

scholarly journals Communication Complexity And Linearly Ordered Sets

2015 ◽  
Vol 29 (1) ◽  
pp. 93-117
Author(s):  
Mieczysław Kula ◽  
Małgorzata Serwecińska
Keyword(s):  
Upper Bound ◽  
Open Problem ◽  
Numerical Experiments ◽  
Communication Complexity ◽  
Ordered Sets ◽  
Finite Sets ◽  
The Family ◽  
Linear Lattices

AbstractThe paper is devoted to the communication complexity of lattice operations in linearly ordered finite sets. All well known techniques ([4, Chapter 1]) to determine the communication complexity of the infimum function in linear lattices disappoint, because a gap between the lower and upper bound is equal to O(log2n), where n is the cardinality of the lattice. Therefore our aim will be to investigate the communication complexity of the function more carefully. We consider a family of so called interval protocols and we construct the interval protocols for the infimum. We prove that the constructed protocols are optimal in the family of interval protocols. It is still open problem to compute the communication complexity of constructed protocols but the numerical experiments show that their complexity is less than the complexity of known protocols for the infimum function.

scholarly journals A Tight Lower Bound for Convexly Independent Subsets of the Minkowski Sums of Planar Point Sets

10.37236/484 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Ondřej Bílka ◽  
Kevin Buchin ◽  
Radoslav Fulek ◽  
Masashi Kiyomi ◽  
Yoshio Okamoto ◽  
...  
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Minkowski Sum ◽  
Independent Subset ◽  
Point Sets ◽  
Planar Point ◽  
Minkowski Sums

Recently, Eisenbrand, Pach, Rothvoß, and Sopher studied the function $M(m, n)$, which is the largest cardinality of a convexly independent subset of the Minkowski sum of some planar point sets $P$ and $Q$ with $|P| = m$ and $|Q| = n$. They proved that $M(m,n)=O(m^{2/3}n^{2/3}+m+n)$, and asked whether a superlinear lower bound exists for $M(n,n)$. In this note, we show that their upper bound is the best possible apart from constant factors.

scholarly journals On the guaranteed estimates of the area of convex subsets of compacts on a plane

2020 ◽  
Vol 12 (4) ◽  
pp. 112-126
Author(s):  
Владимир Николаевич Ушаков ◽  
Vladimir Ushakov ◽  
Александр Анатольевич Ершов ◽  
Alexandr Ershov
Keyword(s):  
Exact Solution ◽  
Lower Bound ◽  
Convex Hull ◽  
Convex Subset ◽  
Upper Bound ◽  
Compact Set ◽  
Geometric Difference ◽  
Convex Subsets

The paper considers the problem of constructing a convex subset of the largest area in a nonconvex compact on the plane, as well as the problem of constructing a convex subset from which the Hausdorff deviation of the compact is minimal. Since, in the general case, the exact solution of these problems is impossible, the geometric difference between the convex hull of a compact and a circle of a certain radius is proposed as an acceptable replacement for the exact solution. A lower bound for the area of this geometric difference and an upper bound for the Hausdorff deviation from it of a given nonconvex compact set are obtained. As examples, we considered the problem of constructing convex subsets from an alpha-set and a set with a finite Mordell concavity coefficient.

scholarly journals Extrapolation of convex functions

Filomat ◽  
2018 ◽  
Vol 32 (1) ◽  
pp. 127-139
Author(s):  
M. Sababheh
Keyword(s):  
Lower Bound ◽  
Convex Hull ◽  
Upper Bound ◽  
Convex Functions ◽  
Jensen Inequality ◽  
Matlab Code

The idea of the well known Jensen inequality is to interpolate convex functions, by finding an upper bound of the function at a point in the convex hull of predefined points. In this article, we present a counterpart of this inequality by giving a lower bound of the function outside this convex hull. This inequality is then refined by finding as many refining positive terms as we wish. Some applications treating means, integrals and eigenvalues are given in the end. Moreover, we present a MATLAB code that helps generate the parameters appearing in our results.

scholarly journals On Combinatorial Depth Measures

2018 ◽  
Vol 28 (04) ◽  
pp. 381-398
Author(s):  
Stephane Durocher ◽  
Robert Fraser ◽  
Alexandre Leblanc ◽  
Jason Morrison ◽  
Matthew Skala
Keyword(s):  
Convex Hull ◽  
Half Plane ◽  
Combinatorial Complexity ◽  
Query Point ◽  
Combinatorial Characterization ◽  
Simplicial Depth ◽  
Natural Measures

Given a set [Formula: see text] of points and a point [Formula: see text] in the plane, we define a function [Formula: see text] that provides a combinatorial characterization of the multiset of values [Formula: see text], where for each [Formula: see text], [Formula: see text] is the open half-plane determined by [Formula: see text] and [Formula: see text]. We introduce two new natural measures of depth, perihedral depth and eutomic depth, and we show how to express these and the well-known simplicial and Tukey depths concisely in terms of [Formula: see text]. The perihedral and eutomic depths of [Formula: see text] with respect to [Formula: see text] correspond respectively to the number of subsets of [Formula: see text] whose convex hull contains [Formula: see text], and the number of combinatorially distinct bisections of [Formula: see text] determined by a line through [Formula: see text]. We present algorithms to compute the depth of an arbitrary query point in [Formula: see text] time and medians (deepest points) with respect to these depth measures in [Formula: see text] and [Formula: see text] time respectively. For comparison, these results match or slightly improve on the corresponding best-known running times for simplicial depth, whose definition involves similar combinatorial complexity.

scholarly journals ON THE BOUNDEDNESS OF AN ITERATION INVOLVING POINTS ON THE HYPERSPHERE

2012 ◽  
Vol 22 (06) ◽  
pp. 499-515
Author(s):  
THOMAS BINDER ◽  
THOMAS MARTINETZ
Keyword(s):  
Convex Hull ◽  
Upper Bound ◽  
Upper Bounds ◽  
Maximal Length ◽  
Unit Hypersphere ◽  
Finite Set ◽  
Set Of Points

For a finite set of points X on the unit hypersphere in ℝd we consider the iteration ui+1 = ui + χi, where χi is the point of X farthest from ui. Restricting to the case where the origin is contained in the convex hull of X we study the maximal length of ui. We give sharp upper bounds for the length of ui independently of X. Precisely, this upper bound is infinity for d ≥ 3 and [Formula: see text] for d = 2.

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