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Let $f(x)=\sin x$, and $g(x)=\sin x + 1$. Consider a set $S=\{(x,y)| f(x)\leq y \leq g(x), x\in [0,2\pi]\}$. This set $S$ can be considered as "Raceway" My question is finding the shortest path in $S$...

I am trying to compute the nonnegative quantity $$ \underset{y\in\mathbb{S}^{d-1}}{\sup}\int_{0}^{t}(\Vert A(\tau)y\Vert_{1}- \Vert A(\tau)y\Vert_{q})d\tau, \quad 1 < q < \infty $$ where $\...