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, convexsubset, 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.