A point outside unit ball in $l^1(\mathbb R^2)$ and inside unit ball in $l^\infty (\mathbb R^2)$. Is there a $p$ between 1 and infinity such that $||x||_p = 1$? How about in $\mathbb R^n?$
Since it is outside the unit ball in $l^1(\mathbb R^2)$ and inside the unit ball in $l^\infty (\mathbb R^2)$, $l^1$ norm is $\geq 1$ and $l^\infty$ norm is $\leq 1$. But I am not sure how to approach this question. How do I get started?
The unit ball in $\mathbb{R}^2$ with respect to the $\lVert\cdot\rVert_1$-norm is the square whose vertices are $(\pm1,0)$ and $(0,\pm1)$, whereas the unit ball in $\mathbb{R}^2$ with respect to the $\lVert\cdot\rVert_\infty$-norm is the square whose vertices are the numbers of the form $(\pm1,\pm1)$. So take, for instance, the point $(a,a)$, with $a\in\left(\frac1{\sqrt2},1\right)$.