Convexity of a European call price in strike ($C(T,a{K}_1+(1-a)K_2)\le aC(T,K_1)+(1-a)C(T,K_2)$)

For any fixed maturity, a European call option price is convex in its strike because $K\mapsto (s-K{)}^{+}$ is convex for every terminal stock value $s$. Specifically, for $0\le a\le 1$, $(s-a{K}_1-(1-a)K_2{)}^{+}\le a(s-K_1{)}^{+}+(1-a)(s-K_2{)}^{+}$. Multiplication by the positive discount factor and expectation under the pricing measure preserve this inequality. This also explains why a violation of convexity creates a butterfly-spread arbitrage.