MathOverflow

Let $K \subset \mathbb{R}^2$ be a convex body. Define two quantities:
Interior mean distance. Let $X, Y$ be independent and uniformly distributed in $K$. Set$$\Delta(K) \;=\; \mathbb{E}\,\|X - Y\|.$$
Boundary mean distance. Let $X', Y'$ be independent and uniformly distributed on $\partial K$. Set$$\theta(K) \;=\; \mathbb{E}\,\|X' - Y'\|.$$...

It is classical that any Lebesgue integrable function on $\mathbb R^n$ can be written as the pointwise a.e. limit of continuous functions with respect to Lebesgue measure. Does this still hold if we consider a lower dimensional Hausdorff measure?
Question: For which $s < n$, if any, does it hold that any Borel, integrable function can be written as the $\mathcal H^s$–a.e. pointwise limit of continuous functions?...

A Malcev category is a finitely complete category in which every reflexive relation is an equivalence relation. One can show that the dual of every elementary topos is a Malcev category (see Borceux-Bourn, Example 2.2.18). In other words, every elementary...

For arbitrary integral domains, it is shown that the class of minimal order types of the images of the transfinite (i.e., $\mathrm{Ord}$-valued) Euclidean functions is the class of indecomposable ordinals $\omega^\alpha$ with $\alpha\in\mathrm{Ord}$...