向量 p范数的凹凸性证明

Suppose $p<1,p\ne 0$. Show that the function
$$f(x)=(\sum _{i=1}^n{x}_i^p{)}^{\frac{1}{p}}$$
with $domf={\mathbb{R}}_n^{++}$ is concave. This includes as special cases $f(x)=({\sum }_{i=1}^n{x}_i^{\frac{1}{2}}{)}^2$ and the harmonic mean$f(x)=({\sum }_{i=1}^n\frac{1}{x_i}{)}^{-1}$

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显然，当 p>=1时，向量的$L_p$范数是凸的。