Example of a metric.

Question

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.

Answer 1

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 $d(x,y) =d_S\bigg((x,f(x)),(y,f(y)) \bigg)$.