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

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.

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.