# Points between the unit balls in $\ell_2^1(\mathbb{R})$ and $\ell_2^\infty(\mathbb{R})$

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.

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

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.