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2 votes

Indecomposable integral currents

I think the following might be an example, though it will require a bit of work if you want to make it more precise: Take an immersion of a sphere, which is injective except for one cap at each pole, ...
  • 1,617
1 vote

isoperimetric problems on Alexandrov spaces

The existence follows from theory of currents the same way as for Riemannian manifolds. As far as I know, there are no regularity results for $\partial D$. But look at the proof of Levy--Gromov ...
2 votes
Accepted

Inequality with decreasing rearrangement and non-decreasing function

$\newcommand\la\lambda\newcommand{\R}{\mathbb R}\newcommand{\de}{\delta}$The answer to this question is yes. Indeed, let $h:=f^*/g$. Then $h$ is a nonincreasing function and the inequality in question ...
2 votes

Why do almost all points in the unit interval have Kolmogorov complexity 1?

By the Point-to-Set principle (Lutz & Lutz 2018), for any $\varepsilon$ the set of real numbers with Kolmogorov complexity at most $\varepsilon$ has Hausdorff dimension $\varepsilon$. As $\mathbb{...
  • 3,659
6 votes
Accepted

Inequality with decreasing rearrangement function

No (if $c$ cannot depend on $f^*$ or $g$). Indeed, let $h:=f^*/g$. Then $h$ can be any positive function and the inequality in question can be rewritten as $$lhs:=\Big(\int_0^\infty h(s)^{p'}ds\Big)^{...
20 votes
Accepted

Why do almost all points in the unit interval have Kolmogorov complexity 1?

I'm not an expert on Kolmogorov complexity, but this does seem like a counting argument: for any fixed $\epsilon > 0$, there are only $\sum_{i = 1}^{(1-\epsilon)n} 2^i < 2^{(1-\epsilon)n+1}$ ...
3 votes

Second order differentiability of convex functions

The following argument is used in my recent paper with D. Azagra and A. Cappello (work in progress). The answer is yes. I asked about whether the proof of Theorem 1 can be modified so to cover Theorem ...

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