The L_1 ball in 2D is shaped like a diamond (L_1 is also known as the Manhattan norm). The L_∞ ball is shaped like a square (L_∞ is also known as the supremum norm). They are similar, i.e. have same shape. The L2 ball is shaped like a circle.
Hypothesis: For all p in the interval (1,2), there is q>2 such that the q-ball and the p-ball are similar. One further hypothesis is that this occurs precisely when p,q are Hölder conjugates.
I wasn't sure how to tag this.
$\ell_p^n$ is isometric to $\ell_r^n$ with $1\le p < r\le \infty$ iff $n=1$ or ($n=2$ and $p=1$ and $r=\infty$ and the scalars are real). While known before L. Dor's 1976 paper in the Israel J. Math. 24, 260-268, it can be deduced from Theorem 2.1 in that paper, which classifies when $\ell_s^n$ isometrically embeds into $L_t(0,1)$.
You can use this applet to generate pictures of the curves you're thinking of. Your conjecture seems unlikely because |x|^p + |y|^p = 1 appears to be a smooth curve for p ≥ 2 and have singularities at (1,0), (-1,0), (0,1), (0,-1) for p < 2.
You are looking for isometries between the ell^p and ell^q
Here is a paper describing some methods on how to identify isometries (but the paper is not freely downloadable)
http://www.informaworld.com/smpp/content~content=a746864853~db=all~order=page
