向量 p范数的凹凸性证明

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Suppose $p < 1, p \neq0$. Show that the function
$f(x) = (\sum_{i=1}^nx_i^p)^{\frac1p}$
with $dom f = \R_n^{++}$ is concave. This includes as special cases $f(x) = (\sum_{i=1}^nx_i^{\frac12})^2$ and the harmonic mean $f(x) = (\sum_{i=1}^n\frac1{x_i})^{-1}$

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显然，当 p>=1时，向量的 $L_p$范数是凸的。