/sci/ - Science & Math

ok so i feel like this should be really easy, but i really can't seem to find any formal argument.

Let [math]y,z\in\mathbb{R}^n, x\in relint(conv({x,y})), A:=B_\varepsilon(y), B:=conv(A\cup {z})[/math] with [math]\varepsilon[/math] small enough such that [math]A[/math] does not contain [math]z[/math]. Now i want to show that [math]x\in int(B)[/math].

Pic related is a sketch for [math]n=2[/math].