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2d
answered Fréchet L-Spaces
Apr
11
revised Is $\max_{\|x\|_p=\|y\|_p=1} |\langle x, Ay\rangle|$ equivalent to $\max_{\|x\|_p=|} |\langle x, Ax\rangle|$ for symmetric $A$ & $p\geq 2$?
edited body
Apr
11
revised Is $\max_{\|x\|_p=\|y\|_p=1} |\langle x, Ay\rangle|$ equivalent to $\max_{\|x\|_p=|} |\langle x, Ax\rangle|$ for symmetric $A$ & $p\geq 2$?
edited body
Mar
4
comment Shape whose translated and scaled copies are closed under intersection
Don't know if these count as a shapes but this also holds for single points, lines and line segments.
Mar
3
comment Extreme couplings
Relevant answer here: mathoverflow.net/a/152271/9652
Mar
3
comment Extreme couplings
In ther discrete case the buzzword is "Birkhoff polytope".
Mar
3
comment Completion of spaces of measures w.r.t. weak norms
@PassingThru Ok, thanks for clarifying. As much as I try to love these two volumes of Bogachev, I always struggle with the notation and diversity of concepts in there.
Mar
2
comment Bounds on the curvature of a sequence of convex functions
I don't think that there is some simple control of this ratio. You could have some $f_n$ that approach a function with a non-differentiable kink at the minimum, even with fixed value at the minimum (some scaled and translated version of $\sqrt{x^2+1/n}$).
Mar
2
comment Completion of spaces of measures w.r.t. weak norms
@PassingThru As far as I see, the norm $\|\sigma\|_0^*$ is exactly as Bogachev defines it in §8.10 for the difference of two Borel probability measures with finite first moments. In fact he has another Kantorovich-Rubinshtein norm in §8.3 but I thought that I could go with this setting as well.
Mar
2
comment Completion of spaces of measures w.r.t. weak norms
@TomekKania And does the completion of measure space w.r.t. the Fortet-Mourier norm also has a name or a different characterization in that community? Or would that make a separate question?
Mar
1
comment Completion of spaces of measures w.r.t. weak norms
Thanks! I really forgot to check your book (which I read some years ago) - also the pointer to Lipschitz-free spaces sounds interesting.
Mar
1
accepted Completion of spaces of measures w.r.t. weak norms
Mar
1
asked Completion of spaces of measures w.r.t. weak norms
Feb
26
comment Alternative representation of $C_c(X)$ as inductive limit
Oops, yes $C(X;K_{n+1})$ is what I meant. For question 2 I have no idea.
Feb
26
comment Alternative representation of $C_c(X)$ as inductive limit
I see not much difference in your construction. Let me say it this way: Since $C(K_n)$ isometrically embeds in $C(X;K_{n+1})$ (as you noted) the "original construction" is the strict inductive limit of the $C(X;K_n)$ while your construction is the strict inductive limit of the $C(K;K_{n+1})$…
Feb
24
comment A problem on real valued functions in $\mathbb{R}^2$ with least variation
I doubt that this will be optimal or almost optimal up to a constant: Consider a wide and flat rectangle-like $\Omega$ and boundary values that are close to one in the middle the short edges and zero along the long edges. The proposed approach by scaling introduce long "edges" in the interior of $\Omega$ while $TV$ would favor short edges…
Feb
24
revised A problem on real valued functions in $\mathbb{R}^2$ with least variation
added 21 characters in body
Feb
23
comment an inverse problem related to gaussian integral
@yimin Sounds strange at first sight, that this may well be possible (note that $f_T(x)$ can be an analytic function in $x$ and $T$ and hence, determined if known on a set with some cluster point). More to the OP: I don't know the answer but I vaguely remember that the book The One-Dimensional Heat Equation by Cannon is a great source for such questions.
Feb
22
comment A problem on real valued functions in $\mathbb{R}^2$ with least variation
May work… Anyway, all the best for you undertakings.
Feb
22
comment A problem on real valued functions in $\mathbb{R}^2$ with least variation
Hmm, this may get complicated as sets of 1-dimensional Hausdorff measure can get quite nasty. Also this assumption would not even imply boundedness of $f$… I think the framework is most natural, if you assume that $f$ is $L^1$ on the boundary. Usually, these spaces of differentiable functions are not well suited for variational problems. The framework of weak derivatives is much more handy here.