Lipschitz extension of affine function to whole space

Question

Consider the metric linear space $(R^n, d)$.

My question is whether for every $x_1,x_2 \in R^n$ there exists an affine function $\alpha : R^n \to R$ satisfying

$\alpha(x_1) - \alpha(x_2) = d(x_1, x_2)$,

$|\alpha(x) - \alpha(y)| \leq d(x, y), \forall x,y \in R^n$ (i.e. $\alpha$ is Lipschitz with constant 1).

I have thought about applying Kirszbraun theorem, but it only guarantees the existence of a Lipschitz function, which is not necessarliy affine.

Answer 1

Fix $x_1 \not= x_2$.

Define $z = \dfrac{x_1 - x_2}{d(x_1,x_2)}$. Clearly $|z| = 1$.

The function $\alpha(x) = z^T \cdot x$ should do the job.