向量 p范数的凹凸性证明

Suppose $p &lt; 1, p \neq0$p<1,p̸​=0. Show that the function
$f(x) = (\sum_{i=1}^nx_i^p)^{\frac1p}$f(x)=(i=1∑n​xip​)p1​
with $dom f = \R_n^{++}$domf=Rn++​ is concave. This includes as special cases $f(x) = (\sum_{i=1}^nx_i^{\frac12})^2$f(x)=(∑i=1n​xi21​​)2 and the harmonic mean$f(x) = (\sum_{i=1}^n\frac1{x_i})^{-1}$f(x)=(∑i=1n​xi​1​)−1

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显然，当 p>=1时，向量的$L_p$Lp​范数是凸的。