18,746 reputation
461144
bio website math.psu.edu/petrunin
location Penn State
age 47
visits member for 5 years, 9 months
seen 9 hours ago

A description by my son:

Mein Vater hat keine Haare aber er hat einen schwarzen Bart. Er hat grüne Augen so wie ich. Er benutzt seine Brille sehr selten. Sein Name ist Tosha. Er mag Musik spielen aber ist ganz schlecht dabei und ist 44 Jahre alt.


Jul
24
comment Two geodesics with angle $\pi$ in Alexandrov space
@RichardMontgomery, it works for any $\beta>0$, but the angle in this case is $<\pi$...
Jul
24
comment Two geodesics with angle $\pi$ in Alexandrov space
For $\phi(t)=|t|$ the angle is $<\pi$, something like $\phi(t)=|t|^{1.00001}$ should work.
Jul
24
answered Two geodesics with angle $\pi$ in Alexandrov space
Jul
20
awarded  Famous Question
Jul
20
comment How to change the given metric if we want to add few extra isometries?
Assume $X=\mathbb R$ and $G=\mathbb R\backslash \{0\}$ acting by muliplication. If an intrinsic metric is invariant with respect to $G$ then both $\mathbb R_\pm$ are isometric to $\mathbb R$. That is the only choice and it does not give an intrisic metric on entire $\mathbb R$.
Jul
19
comment Products of elliptic isometries
The condition $\mathrm{Fix}(g)\cap\mathrm{Fix}(h)=\emptyset$ should be exchanged to something like $$2{\cdot}|{\rm Fix}(g)-{\rm Fix}(h)|_X\approx|{\rm Fix}(g)-h{\cdot}{\rm Fix}(g)|_X.$$ (I do not know a ref.)
Jul
15
comment Why is a negatively curved cone surface locally CAT(-1)?
@HuipingPan, I gave a ref, and yet for sure you have to assume that curvature $\le -1$.
Jul
14
comment Why is a negatively curved cone surface locally CAT(-1)?
Note that if you cut a triangle into two thin triangles then it is thin and use it. See 9.3.4 here math.psu.edu/petrunin/papers/alexandrov-geometry
Jul
9
revised bi-Lipschitz gluing
added 267 characters in body
Jul
8
comment The sign of the mean curvature on convex cones in three dimensions
The answer is yes. Essentailly you need to prove convexity of the intersection, say $F$, of your cone with the unit sphere. Note that the mean curvature of the cone can equals to the curvature of $\partial F$. Then you can proceed the same way as in the plane.
Jul
3
comment How to prove the existence of the polytope in $\mathbb{R}^d$ with a given number of faces, minimizing the isoperimetric ratio?
It follows since the space of configurations of $n$ unit vectors is compact.
Jul
2
answered When is the boundary of an open planar set a Jordan curve?
Jun
27
awarded  Notable Question
Jun
11
comment Gauss-Bonnet formula for 2-dimensional Alexandrov spaces
Yes, GH-continuity is missing in the ref, but it follows easily.
Jun
9
comment How misleading is it to regard $\frac{dy}{dx}$ as a fraction?
I do not think you should remind that y is a function of x if it is not necessary; so for me this is a feature!bug.
Jun
5
comment Alexandrov spaces which are not limits of Riemannian manifolds
in (2) you get examples which are compact and which are not compact.
Jun
4
revised Alexandrov spaces which are not limits of Riemannian manifolds
added 47 characters in body
Jun
4
comment Alexandrov spaces which are not limits of Riemannian manifolds
(1) yes always, (2) the construction is local and one can assume $A$ is compact,
Jun
3
awarded  Notable Question
Jun
3
answered Alexandrov spaces which are not limits of Riemannian manifolds