# Maximise $L^q$ norm of a vector, for fixed $L^1$ and fixed $L^p$ norms [closed]

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

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 (Lagrangian multipliers), it can be shown that $x$ can take at most $2$ distinct non-zero values in its co-ordinates. This reduces the problem to a finite number of cases.

My guess is that the distribution $\left( \alpha, \frac{1-\alpha}{n-1}, \frac{1-\alpha}{n-1}, \ldots, \frac{1-\alpha}{n-1} \right)$ which works for $q = \infty$ should also work for all $q > p$. It seems intuitively true and matlab calculations also support this guess. Any pointers/related problems/inequalities are very much welcome.

-

## closed as no longer relevant by Bill Johnson, Mark Meckes, Gjergji Zaimi, Andrés E. Caicedo, Ryan BudneyDec 20 '11 at 21:11

This question is unlikely to help any future visitors; it is only relevant to a small geographic area, a specific moment in time, or an extraordinarily narrow situation that is not generally applicable to the worldwide audience of the internet. For help making this question more broadly applicable, visit the help center.If this question can be reworded to fit the rules in the help center, please edit the question.

You will find an answer to a more general question here artofproblemsolving.com/Forum/… – Gjergji Zaimi Dec 19 '11 at 10:49
A very similar question is considered in: arxiv.org/pdf/math/9910093 – Igor Rivin Dec 19 '11 at 12:36
This is a nice question but rather elementary for MO. Since the OP has been pointed in the right direction, I vote to close. – Bill Johnson Dec 19 '11 at 16:07
This is exactly what I had been searching for. Thanks! Just for the sake of completion, link to the proof of Equal variable theorem: emis.de/journals/JIPAM/images/059_06_JIPAM/059_06.pdf – VSJ Dec 20 '11 at 9:20