Stability Refinements of the Triangle Inequality in $L^p$ Spaces

Let $(X,\mu )$ be a measure space and let $1<p<\mathrm{\infty }$. We study quantitative stability refinements of Minkowski’s inequality $$\mathrm{\parallel }f+g{\mathrm{\parallel }}_p\le \mathrm{\parallel }f{\mathrm{\parallel }}_p+\mathrm{\parallel }g{\mathrm{\parallel }}_p$$ for real-valued functions in $L^p(X,\mu )$. We first establish a stability estimate for arbitrary real-valued functions and show that its constant is sharp. We then prove that, for nonnegative functions, the constant can be improved when $p\ge 2$, again to its optimal value. More precisely, if $f,g\ge 0$ and $f,g\mathrm{\ne }0$, then $$\mathrm{\parallel }f+g{\mathrm{\parallel }}_p\le \mathrm{\parallel }f{\mathrm{\parallel }}_p+\mathrm{\parallel }g{\mathrm{\parallel }}_p-c_p\min \{\mathrm{\parallel }f{\mathrm{\parallel }}_p,\mathrm{\parallel }g{\mathrm{\parallel }}_p\}{\parallel \frac{f}{\mathrm{\parallel }f{\mathrm{\parallel }}_p}-\frac{g}{\mathrm{\parallel }g{\mathrm{\parallel }}_p}\parallel }_p^{{\alpha }_p},$$ where $$c_p=\{\begin{array}{ll}{\displaystyle \frac{p-1}{4}},& 1<p\le 2,\\ {\displaystyle \frac{1-2^{1-p}}{p}},& 2\le p<\mathrm{\infty },\end{array}{\alpha }_p=\{\begin{array}{ll}2,& 1<p\le 2,\\ p,& 2\le p<\mathrm{\infty }\mathrm{.}\end{array}$$ Both constants are best possible.