It is indeed. The price of an American option is the Bermuda option in the limit that the exercising interval approaches zero. The Bermuda option at any exercising time can be evaluated inductively via the dynamic programming principle as the maximum of the payoff and the risk-neutral expected value of the Bermuda option price at the next exercise time. The latter is inductively assumed to be whilst the former is convex in the strike. The maximum of convex functions is again convex. The dominant convergence theorem guarantees the pointwise limit of a sequence of convex functions is again convex. Therefore the American option is convex in strike. As a matter of fact the same deduction applies to an option where the principle of dynamic programming is applicable any the payoff function is convex with respect to a parameter.