Efficiently reversing the triangle inequality with additional information

Question

Suppose $f$ and $g$ are bounded functions, having whatever niceness properties you want, on some space of finite measure. Assume they are normalized so that $\int |f|^2=\int|g|^2=1$. I am looking for sufficient conditions so that $$\int|f+g|\geq c(\int|f|+\int|g|)$$ for some positive $c$ bounded away from zero. The sort of thing I have in mind is, say, control over $\int |f|^p|g|^p$ for $p>1$, which I interpret as meaning $f$ and $g$ cannot be too correlated. Are there instances where this sort of thing can be done? If not, what do counterexamples look like?

Answer 1

You definitely need certain "niceness" assumptions. For instance, you can look at $f$ and $g$ on $[0,1]$ with $$f(x)=\begin{cases} N & \text{ if }0\leq x\leq \frac{1}{2N^2}\\ \frac{1}{\sqrt{2}} &\text{ otherwise}\end{cases}$$

and $$g(x)=\begin{cases} N & \text{ if }1-\frac{1}{2N^2}\leq x\leq 1\\ \frac{-1}{\sqrt{2}} &\text{ otherwise}.\end{cases}$$

Both of these functions have $L^2$ norm about 1, $L^1$ norm about $1/\sqrt 2$, and $\int|f+g|$ is about $1/N$. Because their spikes are disjointly supported, $$\int |f(x)|^p|g(x)|^p\leq \frac{1}{2^p}+N^{p-2}$$ which is pretty small. By comparison, if $f$ and $g$ had the same spikes, then this would be about$N^{2p-2}$ in magnitude.

Answer 2

Let $(\Omega,\mu)$ be a measure space and let $f,g:\Omega\mapsto\mathbb{R}$ be two bounded real valued functions defined on the space $\Omega$ with $\mu(\Omega)<+\infty$. Since $f$ and $g$ are bounded then there exists $M,N\in\mathbb{R}$ such that $|f|\leqslant M$ and $|g|\leqslant N$ on $\Omega$. Therefore $$\int_{\Omega}|f|+|g|\,d\mu\leqslant \int_{\Omega}(M+N)\,d\mu=(M+N)\mu(\Omega)<+\infty$$ On the other hand from triangle inequality we have $$\int_{\Omega}|f+g|\,d\mu\leqslant\int_{\Omega}|f|\,d\mu+\int_{\Omega}|g|\,d\mu$$ Therefore a sufficient condition would be:

There exists some positive constant $c\leqslant 1$ such that $|f+g|\geqslant c(M+N)$.