Colourful Simplicial Depth
نویسندگان
چکیده
Inspired by Bárány’s Colourful Carathéodory Theorem [4], we introduce a colourful generalization of Liu’s simplicial depth [13]. We prove a parity property and conjecture that the minimum colourful simplicial depth of any core point in any d-dimensional configuration is d2 +1 and that the maximum is dd+1 +1. We exhibit configurations attaining each of these depths, and apply our results to the problem of bounding monochrome (non-colourful) simplicial depth.
منابع مشابه
Advanced Optimization Laboratory Title : Colourful Simplicial
Inspired by Bárány’s colourful Carathéodory theorem [Bár82], we introduce a colourful generalization of Liu’s simplicial depth [Liu90]. We prove a parity property and conjecture that the minimum colourful simplicial depth of any core point in any ddimensional configuration is d + 1 and that the maximum is d + 1. We exhibit configurations attaining each of these depths, and apply our results to ...
متن کاملAlgorithms for Colourful Simplicial Depth and Medians in the Plane
The colourful simplicial depth (CSD) of a point x ∈ R relative to a configuration P = (P , P , . . . , P ) of n points in k colour classes is exactly the number of closed simplices (triangles) with vertices from 3 different colour classes that contain x in their convex hull. We consider the problems of efficiently computing the colourful simplicial depth of a point x, and of finding a point in ...
متن کاملSmall Octahedral Systems
We consider set systems that satisfy a certain octahedral parity property. Such systems arise when studying the colourful simplices formed by configurations of points of in R; configurations of low colourful simplicial depth correspond to systems with small cardinality. This construction can be used to find lower bounds computationally for the minimum colourful simplicial depth of a configurati...
متن کاملA combinatorial approach to colourful simplicial depth
The colourful simplicial depth conjecture states that any point in the convex hull of each of d+1 sets, or colours, of d+1 points in general position inRd is contained in at least d 2 +1 simplices with one vertex from each set. We verify the conjecture in dimension 4 and strengthen the known lower bounds in higher dimensions. These results are obtained using a combinatorial generalization of co...
متن کاملA Note on Lower Bounds for Colourful Simplicial Depth
The colourful simplicial depth problem in dimension d is to find a configuration of (d+1) sets of (d+1) points such that the origin is contained in the convex hull of each set, or colour, but contained in a minimal number of colourful simplices generated by taking one point from each set. A construction attaining d + 1 simplices is known, and is conjectured to be minimal. This has been confirme...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید
ثبت ناماگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید
ورودعنوان ژورنال:
- Discrete & Computational Geometry
دوره 35 شماره
صفحات -
تاریخ انتشار 2006