Suppose p < 1 , p ≠ 0 p < 1, p \neq0 p<1,p̸=0. Show that the function f ( x ) = ( ∑ i = 1 n x i p ) 1 p f(x) = (\sum_{i=1}^nx_i^p)^{\frac1p} f(x)=(i=1∑nxip)p1 with d o m f = R n + + dom f = \R_n^{++} domf=Rn++ is concave. This includes as special cases f ( x ) = ( ∑ i = 1 n x i 1 2 ) 2 f(x) = (\sum_{i=1}^nx_i^{\frac12})^2 f(x)=(∑i=1nxi21)2 and the harmonic mean f ( x ) = ( ∑ i = 1 n 1 x i ) − 1 f(x) = (\sum_{i=1}^n\frac1{x_i})^{-1} f(x)=(∑i=1nxi1)−1
显然,当 p>=1时,向量的 L p L_p Lp范数是凸的。