Talk:Hadwiger conjecture (combinatorial geometry)
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The conjecture
editThe 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)
- 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)
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)
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)
Zonotopes
editIn 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)