We obtain sufficient conditions on an M-embedded or L-embedded space so that every nonempty relatively weakly open subset of its unit ball has norm diameter 2. We prove that, up to renorming, this holds for every Banach space containing c 0 c_0 and, as a consequence, for every proper M-ideal. The result obtained for L-embedded spaces can be applied to show that the above property is satisfied for every predual of an atomless real JBW*-triple. As a consequence, a characterization of the Radon-Nikodym property is obtained in this setting, showing that a predual of a real JBW*-triple E verifies the Radon-Nikodym property if, and only if, E is the l ∞ l_{\infty } -sum of real type I triple factors.