Convexity of a certain function connected to the norm

Question

Suppose that we are given two vectors $x,y$ in a normed space $X$. Can we prove in general that the function $$t\mapsto \|x-ty\|$$ is convex?

It is certainly the case if the normed space has dimension 1.

EDIT: It is convex. One uses convexity of $|\langle x^*, x-ty\rangle|$ and passes to the supremum with $\|x^*\|\leqslant 1$.

Answer 1

The claim follows from the triangle inequality and positive homogeneity of the norm: Take $t_1,t_2\in\mathbb R$, $\lambda\in (0,1)$. Then $$ \|x-(\lambda t_1 + (1-\lambda)t_2)y \| = \| \lambda(x-t_1y) +(1-\lambda) (x-t_2 y)\| \le \lambda \|x-t_1y\| + (1-\lambda) \|x-t_2 y\| $$