Suppose that $c_{1}, c_{2},$ and $c_{3}$ are the prices of European call options with strike prices $K_{1}$ $K_{2},$ and $K_{3},$ respectively, where $K_{3} > K_{2} > K_{\mathrm{L}}$ and $K_{3}-K_{2}=K_{2}-K_{1} .$ All options have the same maturity. Show that
$$c_{2} \leqslant 0.5\left(c_{1}+c_{3}\right)$$
(Hint: Consider a portfolio that is long one option with strike price $K_{1}$, long one option with strike price $K_{3}$, and short two options with strike price $K_{2}$.