Finding bounds of $D_p$-function

Question

I am trying to find the upper and lower bounds of the $D_p$ function, defined as: $$ D_p(x) = \left( \sum_{i=1}^nx_i^p \right)^{1/p} $$ for $0<p<1$ and $x \in \{x=(x_1,..,x_n) | x_i \ge 0, \sum_{i=1}^n x_i =1\} $. Is there a simpler way of doing this than setting up the lagrangian?

Answer 1

$D_p(x)$ is concave by the reverse Minkowski inequality. Your constraint is exactly the set of $x_i$ on a simplex.

It's really not so bad with a Lagrangian. For starters, you can use $D_p^p(x)$ instead. Then the lagrange equations read:

$$px_i^{p-1}=\lambda,$$

giving $x_i=(\lambda/p)^{1/p}$ (remember $x_i\geq 0$), so summing this gives $n(\lambda/p)^{1/p}=1$ or $\lambda=\frac{p}{n^{p}}$ and $x=1/n$. You can check that this is a maximum.

For the minimum, think about $x_1=1$ and $x_i=0$ for all other $i$.