Frank Morgan » Blog Archive » More Solutions to Fermat with Fractional Exponents

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By contrast, Erhard Schmidt’s proof that was published in 1948 uses Schwarz

symmetrization and is much more cumbersome.

http://onlinelibrary.wiley.com/wol1/doi/10.1002/mana.19480010202/abstract ]]>

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a two-suited hand. Also, if the bidding had gone as you stated,

your partner would have been declarer! ]]>

Do our walks ever really lead back to the same spot we started, or is it just a comforting mirage that we journey back toward? Does one need to say goodbye to be free to walk if one has never really returned before? Perhaps one can walk free simply when the mirage fades, when one appears in each moment reborn to their father, mother, brother, sister, wife, child, friends, and self.

Was very nice to meet you in Granada, thanks for sharing so many great stories and the lovely views at the Carmen!

]]>After number theory, Math and Literature was my favorite class at MIT.

]]>long time! I hope all is well.

I just came across this old post, and wanted to remind the readers of an old argument of Buser (“A note on the isoperimetric constant”, ASENS 1982 – Lemma 3.4), for proving the continuity, and in fact the Holder continuity with exponent (n-1)/ n , of the isoperimetric profile. I adapted his argument (in fact, an adaption of Gallot’s adaption of Buser’s argument) in Lemma 6.9 of my paper http://arxiv.org/pdf/0712.4092v5.pdf to the weighted Riemannian setting. The result is as follows:

Let (M,g) denote an n-dimensional (n≥2) smooth complete oriented connected Riemannian manifold (no compactness assumed) and let d denote the induced geodesic distance. Let \mu denote an absolutely continuous probability measure with respect to vol_M, such that its density is bounded from above on every ball (but not necessarily from below, nor do we assume it is continuous). Then the isoperimetric profile I=I(M,d,μ) is absolutely continuous on [0,1], and in fact is locally of H ̈older exponent (n−1)/n.

I guess the key point is that the measure I considered was always a probability (or finite) measure. So it is not the compactness that is essential for the continuity of the profile, but finiteness of the measure (+ local boundedness of the density). Incidentally, I used the Minkowski definition of weighted perimeter, but by regularity theory, this a-poteriori coincides with any other definition used, and in any case is irrelevant for the proof of the above statement.

There are some other useful statements in Section 6 of my paper above, like addressing when is I symmetric about the point 1/2, i.e. I(t) = I(1-t) ? This hold as soon as I is lower-semi-continuous (which is of course typically the case).

All the best,

Emanuel.