Are circles also squares in $(\mathbb{R}^2,||\cdot||_{\infty})$?

Question

In $(\mathbb{R}^2,||\cdot||_{\infty})$ circles appear to be squares. But do squares exist in general normed space when we do not have an inner product, hence no natural notion of orthogonality and right angles?

Answer 1

Let $V$ be a real or complex inner product space with inner product $( \cdot, \cdot)$ and induced norm $||v||=(v,v)^{1/2}.$ Then we have for $v,w \in V$:

$ (v,w)=0 \iff ||v|| \le ||v +tw ||$ for all scalars $t$.

This motivates the following definition:

let $(X, ||\cdot||)$ be a normed space. For $x,y \in X$ define

$ x \perp y : \iff ||x|| \le ||x+ty||$ for all scalars $t$.