### 50 years sets with positive reach -- a survey.

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We show that a Pettis integrable function from a closed interval to a Banach space is Henstock-Kurzweil integrable. This result can be considered as a continuous version of the celebrated Orlicz-Pettis theorem concerning series in Banach spaces.

Let m,n be positive integers such that m < n and let G(n,m) be the Grassmann manifold of all m-dimensional subspaces of ℝⁿ. For V ∈ G(n,m) let ${\pi}_{V}$ denote the orthogonal projection from ℝⁿ onto V. The following characterization of purely unrectifiable sets holds. Let A be an ${}^{m}$-measurable subset of ℝⁿ with ${}^{m}\left(A\right)<\infty $. Then A is purely m-unrectifiable if and only if there exists a null subset Z of the universal bundle $(V,v)\left|V\in G\right(n,m),v\in V$ such that, for all P ∈ A, one has ${}^{m(n-m)}\left(V\in G(n,m)\left|\right(V,{\pi}_{V}\left(P\right))\in Z\right)>0$. One can replace “for all P ∈ A” by “for ${}^{m}$-a.e. P ∈...

We give a non-probabilistic proof of a theorem of Naor and Neiman that asserts that if (E, d) is a doubling metric space, there is an integer N > 0, depending only on the metric doubling constant, such that for each exponent α ∈ (1/2; 1), one can find a bilipschitz mapping F = (E; dα ) ⃗ ℝ RN.

We state and prove a stability result for the anisotropic version of the isoperimetric inequality. Namely if $E$ is a set with small anisotropic isoperimetric deficit, then $E$ is “close” to the Wulff shape set.

We introduce and study a rough (approximate) curvature-dimension condition for metric measure spaces, applicable especially in the framework of discrete spaces and graphs. This condition extends the one introduced by Karl-Theodor Sturm, in his 2006 article On the geometry of metric measure spaces II, to a larger class of (possibly non-geodesic) metric measure spaces. The rough curvature-dimension condition is stable under an appropriate notion of convergence, and stable under discretizations as...

We find a condition for a Borel mapping $f:{\mathbb{R}}^{m}\to {\mathbb{R}}^{n}$ which implies that the Hausdorff dimension of ${f}^{-1}\left(y\right)$ is less than or equal to m-n for almost all $y\in {\mathbb{R}}^{n}$.