# Example of a metric.

I am looking for an example of a metric $$d(x,y)$$ on a vector space $$X$$ such that is neither a discrete metric nor induced by a norm and satisfies: $$d(0,ax+(1-a)y)\leq {ad(0,x)}+(1-a)d(0,y),\ \forall x,y \in X, \ \forall a\in[0,1]$$ But I could not find or construct. Please give some hint or any example.

• The inequality resembles a concave function very much. Maybe you can look into that? – thedilated Feb 8 at 5:10

When $$x=(x_1,x_2)\in X = \mathbb{R}^2$$, then $$f : X\rightarrow \mathbb{R},\ f(x)=x_1^2-x_2^2$$.
Then graph of $$f$$ is a saddle surface $$(S,d_S)$$ in $$\mathbb{E}^3$$ where $$d_S$$ is a intrinsic metric. Here $$d(x,y) =d_S\bigg((x,f(x)),(y,f(y)) \bigg)$$.