23,786 reputation
56117
bio website pdmi.ras.ru/~svivanov
location Saint Petersburg, Russia
age 42
visits member for 4 years, 9 months
seen 3 hours ago
I am a research fellow at St.Petersburg Department of Steklov Math Institute

Sep
17
awarded  Enlightened
Sep
17
awarded  Nice Answer
Sep
6
awarded  Enlightened
Sep
6
awarded  Nice Answer
Jul
2
awarded  Curious
Apr
28
awarded  Good Answer
Mar
29
awarded  Popular Question
Mar
3
awarded  Yearling
Feb
24
awarded  gn.general-topology
Feb
24
awarded  Enlightened
Feb
23
awarded  Nice Answer
Feb
23
answered Is a left topological group which is a manifold a topological group?
Feb
17
comment Characterization of discs
There are still counter-examples with the formulation "open set homeomorphic to the disc". For example, start with the unit disc and remove a segment between a point in the boundary and a point inside. You can then remove a small disc centered at the interior endpoint, and so on. In these examples, the boundary contains the unit circle so the property holds as stated. You need to assume that there are no other points on $L\cap\partial D$.
Feb
17
comment A Converse to the Gauss Bonnet Theorem
For example, contract a hemisphere to a segment by contracting each radial circle to a point. The resulting space is a bouquet of a sphere and a circle. Place the segment to the sphere by an injective map. This can be made smooth.
Feb
14
comment A Converse to the Gauss Bonnet Theorem
Take a point with the highest $z$-coordinate. If it is on the boundary, take one with the lowest $z$-coordinate. Since the entire boundary is on the same level, either a highest or lowest point is in the interior.
Feb
12
answered A Converse to the Gauss Bonnet Theorem
Jan
2
awarded  Nice Question
Dec
14
comment Example of non-closed convex hull in a CAT(0) space
Thanks, now I understand. Indeed the Euclidean intuition fails.
Dec
13
comment Example of non-closed convex hull in a CAT(0) space
@Anton, I don't get it. In any manifold, take a point $p$ and a nearby small ball $B$, and connect $p$ to all points of $B$ by geodesic segments. I believe the resulting set is convex if everything is small enough, yet the conical part of the boundary is filled by geodesics.
Dec
9
comment Tverberg's theorem in CAT(0) spaces
Answering my previous comment: see Theorem 1.1 in arXiv:1011.1802 for $r=2$. (Thanks to Roman Karasev who informed me about this reference.)