bio  website  math.psu.edu/petrunin 

location  Penn State  
age  46  
visits  member for  5 years, 5 months 
seen  3 mins ago  
stats  profile views  11,248 
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.
1m

revised 
Are there smooth bodies of constant width?
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Apr 12 
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Gradient estimate of convex functions
$g$ is piecewise linear (not piecewise constant). 
Apr 12 
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Gradient estimate of convex functions
You can find a counterexample for $d=1$ among functions which are linear on each interval $[n,n+1]$. 
Apr 12 
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Existence of shortest paths in complete Alexandrov spaces
Well, this is true for any locally compact lenght metric space (see HopfRinow theorem). You need to show that finite dimensional Alexandrov spaces are locally compact. The later is proved already in BuragoGromovPerelman. 
Apr 12 
asked  biLipschitz gluing 
Apr 6 
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Terminology for metrics?
In this case, you can assume that $C=1$. In this case it is called $E$isometry. Usually it is assumed that $E$ is small, but one does not have to do that. The identity map $(X,d)\to (X,\delta)$ is also called $E$Hausdorff approximation. Hope it helps. 
Apr 5 
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Smoothing operator raising the smoothness exactly by one
@IgorBelegradek, I think you know what I mean, do not you? 
Apr 5 
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Smoothing operator raising the smoothness exactly by one
@IgorBelegradek corrected. 
Apr 5 
revised 
Smoothing operator raising the smoothness exactly by one
added 89 characters in body 
Apr 4 
answered  Smoothing operator raising the smoothness exactly by one 
Apr 4 
comment 
Smoothing operator raising the smoothness exactly by one
@IgorBelegradek, obviously I meant $$\varepsilon\cdot\int\limits_0^x(f(t)\bar f)\cdot dt.$$ 
Apr 4 
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Smoothing operator raising the smoothness exactly by one
For $M=\mathbb S^1$ one can take $C^\infty$smoothing and add $\varepsilon\cdot\int (f(x)\bar f)\cdot dx$, where $\bar f$ is the average value of $f$. 
Apr 3 
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convex decompositions of the sphere
Yes, and hopefully the formula is correct. 
Apr 3 
answered  convex decompositions of the sphere 
Mar 31 
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Generalize GaussBonnet Formula to nonsimple closed curves
@AlexDegtyarev, you are right, "it is freshmen differential geometry", but you wrong about right hand side. 
Mar 28 
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Lower boundedness of the Ricci curvature
@AlexM. it simply impossible to give an answer here, one has to write 20 pages or so and yes Gromov's "Sign and geometric meaning of curvature" is a good sours. 
Mar 27 
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Lower boundedness of the Ricci curvature
Way too vague for me. 
Mar 24 
answered  distributional Hessian for semiconvex functions on nonsmooth manifolds 
Mar 14 
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Are there CAT(1) spaces which are not trees whose Gromov boundary is disconnected?
word hyperbolic groups are not exactly CAT(1). (But there are many examples.) 
Mar 14 
answered  Are there CAT(1) spaces which are not trees whose Gromov boundary is disconnected? 