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Consider a vector $x \in \mathbb R_+^n$ and $p,q \in \mathbb R$ such that $1.

We fix $\sum \limits_{i=1}^{n}|x_i| = 1$ and $ \left(\sum \limits_{i=1}^{n}|x_i|^p \right)^\frac{1}{p} = \theta < 1$.

In other words, $x$ is a probability mass function, with a fixed $p$-norm.

The question is to find that $x$ which maximises the $q$-norm i.e. maximises $\left( \sum \limits_{i=1}^{n}|x_i|^q \right)^\frac{1}{q}$.

Some observations:

  1. If $q = \infty$, the problem reduces to finding that $x$ which has the largest value for $\sup \limits_i \; x(i)$ leading to the distribution which looks like $\left( \alpha, \frac{1-\alpha}{n-1}, \frac{1-\alpha}{n-1}, \ldots, \frac{1-\alpha}{n-1} \right)$.

  2. Using perturbation arguments, it can be shown that $x$ can take at most $2$ distinct values in its co-ordinates (or one of the co-ordinates is $0$, which just reduces the dimension of the problem).

My guess is that the $x$ which works for $q = \infty$ should also work for all $q > p$.

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    I'm still stuck at the same point.. but atleast matlab calculations haven't yielded any counter-examples (i.e the optimal at $q = \infty$ is optimal throughout). It seems cute enough to permit an analytical solution.2011-12-17

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After searching around and asking on mathoverflow, it turns out there is a theorem specifically tailored for this type of norm-constrained problems. Its called the Equal variable theorem and is proved here: http://www.emis.de/journals/JIPAM/images/059_06_JIPAM/059_06.pdf