$u:\mathbb R^n\to\mathbb R$ is a quasi-concave utility function so the indifference curves are convex.
$a,b\in\mathbb R^n$ are two points. Our budget set is the (one-dimensional) segment $[a,b]$ that connects $a$ and $b$.
Let $b'$ be a point in the segment $[a,x^*]$. That is: $b'=\lambda a+(1-\lambda)x^*$ for any $\lambda\in[0,1]$.
Graphically this result is very straight forward but I don't know how to mathematically prove it.
I think we could start of proving that $u(\lambda a+(1-\lambda) x^*)$ is monotonically decreasing with $\lambda$.
Are there named theory related?