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Suppose that you have a bounded function $f(x)$ on a compact domain in $\mathbb{R}^n$. It's easy to see from Holder's inequality that $$ ||f||_1 \leq \operatorname{Volume}(D) ||f||_\infty. $$ There ...

Assumptions Let $L_p(x,y)=(\sum_i|x_i - y_i|^p)^{1/p}$ (Minkowski metric), $a,b$ be arbitrary $n$-dimensional points, $c$ be a point that satisfies $L_p(a,b) = L_p(a,c) + L_p(c,b)$, i.e., a point ...

For which functions $f:\mathbb{R}^+ \to \mathbb{R}^+$ does the inequality $f^{-1}\left(\sum\limits_{k=1}^n f(x_k+y_k)\right) \leq f^{-1}\left(\sum\limits_{k=1}^n f(x_k)\right) + f^{-1}\left(\sum\...