Use induction to show that a series is convex.

Question

I'm having trouble with this:

Use induction to show that $f(x) = \sum\limits_{i=1}^nr_i(x-x_i)^2$ is convex on $\mathbb{R}$ where $\{r_1, \dots, r_n\}$ are positive numbers and $\{x_1, \dots, x_n\}$ are fixed points.

I really don't know where to start or go, any help is appreciated.

Answer 1

Convexity of $r_1(x-x_1)^2\,$: Notice that the second derivative of this is $2r_1$ which is positive. Hence is strictly convex (reference).

Now induction hypothesis being: $n$ terms summation is convex. Need to prove $n+1$ terms summation is convex. Refer "Prove that the sum of convex functions is again convex." for the proof.