Tapio Rajala
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 Dec 20 awarded Yearling Mar 28 comment How can dimension depend on the point? Indeed, it was. Mar 28 revised How can dimension depend on the point? added 1 character in body Mar 28 answered How can dimension depend on the point? Mar 28 comment How can dimension depend on the point? Doesn't upper semicontinuity follow directly from the definition? Dec 20 awarded Yearling Oct 8 comment Interior cone condition preserved on a small perturbation of the domain. @BeniBogosel With the definition you gave, the question of the OP should have a quite immediate positive answer. This is because having a fixed direction for the cone inside the ball says that the boundary is a graph of a Lipschitz function there. The boundedness of $\Omega$ says that the boundary is compact and hence a finite union of Lipschitz graphs. Aug 24 comment A family Mersenne composite numbers? @joro No typo there. Any prime factor is of the form $k*2m+1$. I calculated $2^m \mod p$ with all primes of that form for $k$ up to $2^{28}$. Aug 24 comment A family Mersenne composite numbers? I did a tiny amount of trial factoring: any prime factor of $2^m-1$ with $t=71$ is larger than $2^{172}$. Aug 24 comment A family Mersenne composite numbers? $4*2^w+w$ is prime (or at least prp) for $w=141,411,5495,6647,7427,7889,14565,17933,...$ Aug 22 comment boundary density of the Von Koch flake If you consider the Jordan curve case and assume that the iterated function system that produces the boundary (locally) satisfies the open set condition, then the boundary has positive Hausdorff measure with the similitude dimension and the iterated open sets in the OSC contain at least a fixed portion (in Lebesgue measure) of the set $K$ in comparison to the measure of the open set. Thus for small $r$ the quantity $f_r(x)$ is uniformly bounded below by a positive constant and the integral you consider is positive. Aug 21 answered boundary density of the Von Koch flake Aug 7 answered On a.e. approximate differentiability of certain continuous real functions May 21 comment Can we obtain topology results using analysis in metric measures spaces? The proof for Bonnet-Myers you can find from the second paper by Sturm, for instance. Gigli might visit all such results during the course (depending how long the course is). If you want to read the basic stuff by yourself, I would suggest Villani's book: Optimal transport, old and new. Luckily it is freely available online cedricvillani.org/wp-content/uploads/2012/08/preprint-1.pdf May 21 comment Can we obtain topology results using analysis in metric measures spaces? It is not so easy to answer your question. We do have Bonnet-Myers in CD(K,N) spaces, but here the convexity-type condition is using the Rényi entropy and not the Shannon entropy. May 21 comment Can we obtain topology results using analysis in metric measures spaces? Have you read the papers by Sturm dx.doi.org/10.1007/s11511-006-0002-8 dx.doi.org/10.1007/s11511-006-0003-7 and by Lott and Villani dx.doi.org/10.4007/annals.2009.169.903 ? May 16 answered PDE-Based Triangle Inequality for Optimal Transportation May 15 awarded Popular Question Apr 23 comment Modulus of of continuity of a convolution operator with respect to Wasserstein metric I added a way to see the reverse inequality. I'll think about your followup question later when I have more time. Apr 23 revised Modulus of of continuity of a convolution operator with respect to Wasserstein metric added 489 characters in body