Consider a pair of American and European puts with the same specifications except the former has the continuous early exercise right. Has anyone plotted the Gamma's of both as functions of the underlying price and time to expiry for the underlying greater than the critical exercise price? Is the American put Gamma necessarily greater than or equal to that of the European counterpart in this domain? I would like a mathematical proof if it is true. I suspect the negative answer may predominantly come from the region where the underlying is close to and above the critical exercise price.
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3$\begingroup$ Does this question help? $\endgroup$– KevinCommented May 11, 2023 at 14:48
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$\begingroup$ @Kevin: Yes, it does. Thank you. I am now more interested in a proof if this inequality is always true. $\endgroup$– HansCommented May 11, 2023 at 16:00
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I think the argument of continuity as suggested by the deleted post does apply. American options should be continuous in all their Greeks across “boundaries”, because it is a free boundary. Given continuity , the statement fails. For example the following diagram makes it pretty obvious. Not a math proof I 
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$\begingroup$ This is false. $\frac{\partial^2 P}{\partial S^2}$ is not necessarily continuous across the exercise, or free, boundary. In fact, in most cases it is discontinuous across the boundary. Your diagram just shows the continuity of $\frac{\partial P}{\partial S}$. $\endgroup$– HansCommented May 12, 2023 at 13:24