Product of sets with the Radon-Nikodym Property (RNP)

Question

I have read that it is somewhat well-known that if two Banach spaces $X$ and $Y$ have the Radon-Nikodym Property (RNP), then their product $X\times Y$ also has the RNP.

Does the above result generalize to product of subsets with the RNP? If $C\subset X$ is a (nonempty) bounded, closed, convex subset, then is it true that $$ C\times Y $$ also has the RNP, provided $Y$ has the RNP?

More precisely, I am interested in the product of the form $C\times \Bbb R$, where $C$ is as above. Any result that includes this special case would suffice for my purpose.

Answer 1

In Section 2 of Bourgin's book "Geometric aspects of convex sets with the Radon-Nikodym Property," it is shown that for a closed, convex set non-empty set $K$, having RNP is equivalent to having the martingale convergence property. Assume $C_1\subset X_1$, $C_2\subset X_2$ are closed, convex with RNP. Any martingale $(f_n)_{n=1}^\infty$ taking values in a closed, bounded, convex subset of $C_1\times C_2$ can be written as $f_n=(g_n, h_n)$, where $g_n$ takes values in a closed, bounded, convex subset of $C_1$ and $h_n$ takes values in a closed, bounded, convex subset of $C_2$. Then since $C_1, C_2$ have RNP, they have the martingale convergence property, and $g_n$ and $h_n$ are a.s. convergent to some limits $g,h$ in $L_1(X_1)$ and $L_1(X_2)$, respectively, Then $f_n=(g_n, h_n)$ converges a.s. to $(g,h)\in L_1(X_1\times X_2)$. This shows that $C_1\times C_2$ has the martingale convergence property.