1
$\begingroup$

For $1 \le p \le q \le \infty$, I need an inequality bounding the $\ell_q$ norm from above by the $\ell_p$ norm on $\mathbb{R}^n$: finding a $\lambda$ so that $$ \Vert v \Vert_q \le \lambda \Vert v \Vert_p $$ for all $v$ in a polyhedral cone in the first quadrant. It's easy to see that $\lambda=1$ works without the restriction on $v$ (and is realized by a vector that is zero in all but one place), but that's not good enough for my applications. The polyhedral cone I'm interested in is well inside the interior of the quadrant.

I initially hoped that the maximal value of the ratio $\Vert v\Vert_q/\Vert v\Vert_p$ would be achieved at the vertices of the polyhedral cone, but that is not true when $p > 1$. (If it were true, that estimate would have been good enough for my applications.)

What features of the cone should I be looking at?

My apologies if this question has well-known answers.

Improve this question
$\endgroup$
3
  • $\begingroup$ Let $C$ denote the first quadrant in $\mathbb{R}^2$. I'm not familiar with the concept of polyhedral cones, but if the wikipedia definition is correct then $C$ is a a polyhedral cone. Then $x_k:=(1,\frac{1}{k})$ forms a sequence in $C$ with $\|x_k\|_q/\|x_k\|_p\to 1$, so that $\lambda=1$ is the best possible constant guaranteed in general. Similar examples hold for $n>2$. $\endgroup$
    – Ben W
    Commented Dec 6, 2016 at 14:48
  • $\begingroup$ @BenWallis, yes, the first quadrant is a polyhedral cone. But I have a particular polyhedral cone I'm interested in, contained in a small piece of the first quadrant. The particular cone is too complicated to make sense to describe, I'm more looking for good ways to estimate $\Vert v\Vert_q$ in terms of $\Vert v\Vert_p$. $\endgroup$
    – Dylan Thurston
    Commented Dec 7, 2016 at 0:02
  • $\begingroup$ You can try to use the Mazur map (see Benyamini-Lindenstrauss, Geometric Nonlinear Functional Analysis, volume 1) and map your set on the sphere to $\ell_1$, but it is far from being clear whether this can help. $\endgroup$
    – August Cleaner
    Commented Dec 7, 2016 at 23:55

0

Reset to default

You must log in to answer this question.

Browse other questions tagged .