MathOverflow

Let $h:D\to \mathbb{R}$ be $C^1$, where$D\subset \mathbb{R}^2$ is a connected open set. Let $P,Q\in D$. Given constants $\alpha,\beta,L_0>0$, is there a path parametrized by arclength $t$:$$r:[0,L]\to D,\quad |r'(t)| = 1,\quad r\in C^2$$with $r(...

I have to work with the following variation of the Minkowski sum:
Let $\mathbb E$ be a Euclidean space and $K$ be a convex set of $\mathbb E\times \mathbb E$.Set$$K^+=\{\,x+y\in\mathbb E\mid(x,y)\in K\,\}.$$
Note that if $K=K_x\times K_y$ for some convex sets $K_x$ and $K_y$ in $\mathbb E$, then $K^+$ is the usual Minkowski sum of $K_x$ and $K_y$....

This topic came up in a seminar I was at today, and while those of us in the discussion were able to figure out the approach below, none of us knew any references for it and I could not find it in any of the Riemannian geometry textbooks I have, so I...

I would like to justify rigorously the following local integrability improvement:and if $u\in L^\infty(0,T;L^2(R^N) \cap L^\infty(R^N))\cap L^p(0,T;W^{1,p}(R^N)) \cap L^p(0,T;W^{s,p}(R^N)),\quad\forall T>0 $, can we say $|\nabla u|^{p-2}\in L^1_...