Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Consider the unit ball in $\ell_2^1(\mathbb{R})$ (that is, the unit ball in $\mathbb{R}^2$ with the $d_1$ metric) and the unit ball in $\ell_2^\infty(\mathbb{R})$. I want to show that every point between these balls lies on the boundary of a unit ball in $\ell_2^p(\mathbb{R}^2)$ for $1 < p < \infty$. This seems like a pretty intuitive result but of course I have to prove it.

So far I have that for a point $x_0$ between these unit balls, there exist $a$ and $b$ with $a < b$ such that $x_0$ lies between the $b$ unit ball in $\ell_2^b(\mathbb{R})$ and the $a$ unit ball in $\ell_2^a(\mathbb{R})$. Can I then apply the Intermediate Value Theorem theorem to say that $x_0$ lies on the boundary of a $p$ unit ball for $a < p < b$? That's really the only way I know how to proceed, but any other suggestions would be welcome.

share|improve this question
    
Applying the Intermediate Value Theorem as you suggest seems like a good idea to me. –  Rasmus Nov 8 '11 at 10:58

1 Answer 1

Suggestion: Use the monotonic nature of unit balls. Each ball is the subset or superset of others. Finding least superset ball and greatest subset ball including $x_0$ would perhaps help argument.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.