向量 p范数的凹凸性证明
Suppose
p
<
1
,
p
≠
0
p < 1, p \neq0
p<1,p=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。