702 reputation
27
bio website
location DE
age
visits member for 1 year, 10 months
seen 2 days ago

2d
comment A problem on chains of squares — can one find an easy combinatorial proof?
A) What is the well-known solution that uses Brouwer's fixed-point 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 crossing-free drawing of the (non-planar) 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 non-planarity 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 3-manifolds
fixed grammar and spelling
2d
answered Powers of orthogonal matrices is closed
2d
suggested suggested edit on Simplifying triangulations of 3-manifolds
Jan
20
awarded  Yearling
Oct
22
comment Tiling a rectangle with weighted cells (min-max 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.