Problem Let $0<p<1$, show that the neighbourhoods $\{f \in L^p(0,1):||f||_p<\epsilon\}$ of the zero function are not convex.
I am pretty stuck with this problem. If I've understood the statement of the exercise correctly, for each $\epsilon>0$, I have to show that there are $f,g$ with $||f||_p, ||g||_p <\epsilon$ but with $||\alpha f+(1-\alpha)g||_p \geq \epsilon$ for some $0 <\alpha<1$. I have no idea how to construct this example, I would appreciate hints or suggestions. Thanks in advance.
You could try to find an example in $\mathbb R^2$ first, i.e., $f,g\in \mathbb R^2$ with $|f|_p,|g|_p\le 1$ and $|(f+g)/2|_p>1$. Then use this to construct functions $f,g$.