bio  website  

location  DE  
age  
visits  member for  1 year, 10 months 
seen  2 days ago  
stats  profile views  305 
2d

comment 
A problem on chains of squares — can one find an easy combinatorial proof?
A) What is the wellknown solution that uses Brouwer's fixedpoint theorem? B) The discrete problem has definitely easy direct proofs, just by specializing the polygonal Jordan curve theorem. C) If the two paths don't cross, one can extend them to a crossingfree drawing of the (nonplanar) complete bipartite graph $K_{3,3}$ by adding two new vertices and some curves outside the square. But I guess this does not count, as nonplanarity of $K_{3,3}$ is based on Euler's polyhedral formula, and that in turn depends on the Jordan curve theorem, it seems. 
2d

comment 
Powers of orthogonal matrices is closed
@Sebastian, What do you mean, "any real matrix", for which matrix? Was my argument not short enough? 
2d

revised 
Simplifying triangulations of 3manifolds
fixed grammar and spelling 
2d

answered  Powers of orthogonal matrices is closed 
2d

suggested  suggested edit on Simplifying triangulations of 3manifolds 
Jan 20 
awarded  Yearling 
Oct 22 
comment 
Tiling a rectangle with weighted cells (minmax problem)
What precisely do you mean by "sparse"? A constant number of entries in each row and column? At most $m+n$ entries in total? Or why would the total number of entries not appear in the runtime? (And edit the problem rather than posting an answer) (And the upper bound $a,b<1$ seems to be irrelevant.) 
Oct 11 
awarded  Necromancer 
Oct 11 
answered  Tiling A Rectangle With A Hint of Magic 
Oct 10 
revised 
How many vertices/edges/faces at most for a convex polyhedron that tiles space?
added the original reference of the figure 
Oct 10 
suggested  suggested edit on How many vertices/edges/faces at most for a convex polyhedron that tiles space? 
Oct 8 
revised 
The Cayley Menger Theorem and integer matrices with row sum 2
small clarification 
Oct 8 
comment 
Vertices of a Polytope
Indeed, fullerenes are a good example. Almost all of the surface is a hexagonal lattice, and there you can simultaneously cut away vertices as long as their minimum distance is 2. Thus, you can cut away half of the vertices (almost; staying away from the pentagons). 
Oct 7 
answered  Vertices of a Polytope 
Sep 18 
comment 
Is there a 3d equivalent of this picture?
1. It should be pointed out that the density does not increase indefinitely inside the "black spot" but there is some minimum distance (if the points were really generated as indicated in the text) 2. Would you tolerate, for example (in the planar example) occasional vertices with 5 or 7 incident triangles, or should it be "completely regular"? 
Jun 25 
awarded  Revival 
Jun 25 
awarded  Excavator 
Apr 28 
comment 
What is the expected value for this
The exponent would not change. For example, take random points in a circular disk. Fit a triangle in the disk. A constant fraction of the n points falls inside the triangle (w.h.p.), and thus you get at least $c′n^{1/3}$ points in convex position; similarly, an ellipse (affine image of a disk) fits into any triangle, showing the other direction of the inequality. Any two convex shapes are related like this (after affine transformation). 
Apr 28 
revised 
The Cayley Menger Theorem and integer matrices with row sum 2
corrected info about the sign 
Apr 28 
comment 
The Cayley Menger Theorem and integer matrices with row sum 2
Yes, it is. I have updated my answer. 