Talk:Hadwiger conjecture (combinatorial geometry)

A fact from Hadwiger conjecture (combinatorial geometry) appeared on Wikipedia's Main Page in the Did you know column on 16 March 2009, and was viewed approximately 7,614 times (disclaimer) (check views). The text of the entry was as follows:

Did you know... that the Hadwiger conjecture (diagram pictured) implies that the surface of any three-dimensional convex body can be illuminated by only eight light sources, but the best proven bound is that 16 lights are sufficient?

A record of the entry may be seen at Wikipedia:Recent additions/2009/March.

Wikipedia

The conjecture[edit]

The statement of the conjecture doesn't quite make sense. The text speaks of 2^n - 1 scalars s_i and 2^n - 1 vectors v_i, but in the formula the union runs over 2^n scalars and vectors. Should the union start at 1 rather than 0? That would fix it.

However, please don't edit it unless you know that the correct statement involves 2^n - 1, rather than 2^n, little copies of K. That's why I haven't. 86.1.196.219 (talk) 05:02, 16 March 2009 (UTC)[reply]

Sorry, editing glitch between different versions of the article. A weaker version of the conjecture is that 2^n are needed; a stronger version is that the only case that requires 2^n is a parallelepiped and everything else requires 2^n-1. I'll look into cleaning this up. —David Eppstein (talk) 05:09, 16 March 2009 (UTC)[reply]

The formulation of the equivalent illumination problem does not make sense. Using the current definition of "illuminated", any n+1 floodlights placed at the vertices of a simplex containing the convex body would illuminate it. There is something more complex needed, such as the ray from the floodlight intersecting the interior of the body. Ratfox (talk) 00:55, 27 May 2010 (UTC) Fixed it. The new definition is a bit more cumbersome, but I do not know how to make it simpler... Ratfox (talk) 01:01, 27 May 2010 (UTC)[reply]

Hi, David. The general formula is n >= 2, L(n) = (2^(n-1) -1)^2 *6 -(3*(n-1)^2 -3*(n-1) +1). It took a while to see it, and you'll probably disagree, but it can be found with the first two amounts of data,... 5, 47. Let it be known! William Bouris 2602:306:CF2A:90D0:1C3C:BCDA:E24E:3004 (talk) 22:31, 4 June 2016 (UTC)[reply]

Zonotopes[edit]

In the "Known results" section, it is mentioned that Zonotopes need only (3/4)2^n smaller copies to be covered. Nevertheless, parallelepipeds are zonotopes and they need at least 2^n (this is particularly easy to see for the cube, the rest follows by affine transformation. There is also no citation for it so I would presume this is just not true. I am far from an expert in the field though and therefore reluctant to change the article, but if someone more invested in the area took a look at it and added some missing condition or so, it would be appreciated.Felix Schroeder (talk) 15:20, 21 November 2018 (UTC)[reply]