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It defines the set $H^{p}_{\varepsilon}=\lbrace f \in L^{p}(0,1):\Vert f\Vert _{p}=(\int \vert f\vert^{p}dm)^{\frac{1}{p}}<\varepsilon\rbrace$ with respect to the measuring space $((0,1),\mathbf{B},m)$ I have a great concern about whether when $0<p<1$, then $H^{p}_{\varepsilon}$ is convex? In advance thank you very much for any suggestions

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No, for $0<p<1$ this is not convex. Try a two-dimensional case. If you insist on space $(0,1)$, that means try functions of the form $a$ on $(0,1/2)$ and $b$ on $[1/2,1)$. –  GEdgar Feb 27 '13 at 20:57

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