TITLE: Continuous automorphism groups of normed vector spaces?
QUESTION [11 upvotes]: Consider the metric space on, say, ℝ2 induced by the various $L^p$ norms, and the group of isometries from that space into itself that preserve the origin. When $p=2$ I get the continuous group of rotations, but when $p\in\{1,3,4,5,...\infty\}$ it looks like I just get $D_8$, the symmetry group of the square. Question: what's going on here? Why is 2 so special? Are there other natural norms on ℝ2 (or on ℝn) besides the euclidean one that give interesting isometry groups?

REPLY [3 votes]: Consider the following norm on $\mathbb{R}^{2}$:                                    $||(x,y)||$ := $|x|+|y|$
if $xy\leq0$; $||(x,y)||$ := $|y|$
if $xy\geq0$ and $|y|$ $\geq3|x|$; 
$||(x,y)||$ := $|x|+\frac{2}{3}|y|$
if $xy$ $>0$ and $|y|$ $\leq3|x|$. Then the group of isometries
is { $\pm I\ $}.