= Convexity of a European call price in strike
{title2=$C(T,aK_1+(1-a)K_2)\leq aC(T,K_1)+(1-a)C(T,K_2)$}
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 $0\leq a\leq1$, $(s-aK_1-(1-a)K_2)^+\leq a(s-K_1)^++(1-a)(s-K_2)^+$. 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.
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