This is an announcement for the paper "On the intrinsic and the spatial numerical range" by Miguel Martin, Javier Meri and Rafael Paya. Abstract: For a bounded function $f$ from the unit sphere of a closed subspace $X$ of a Banach space $Y$, we study when the closed convex hull of its spatial numerical range $W(f)$ is equal to its intrinsic numerical range $V(f)$. We show that for every infinite-dimensional Banach space $X$ there is a superspace $Y$ and a bounded linear operator $T:X\longrightarrow Y$ such that $\ecc W(T)\neq V(T)$. We also show that, up to renormig, for every non-reflexive Banach space $Y$, one can find a closed subspace $X$ and a bounded linear operator $T\in L(X,Y)$ such that $\ecc W(T)\neq V(T)$. Finally, we introduce a sufficient condition for the closed convex hull of the spatial numerical range to be equal to the intrinsic numerical range, which we call the Bishop-Phelps-Bollobas property, and which is weaker than the uniform smoothness and the finite-dimensionality. We characterize strong subdifferentiability and uniform smoothness in terms of this property. Archive classification: Functional Analysis Mathematics Subject Classification: 46B20; 47A12 Remarks: 12 pages The source file(s), MartinMeriPaya.tex: 40725 bytes, is(are) stored in gzipped form as 0503076.gz with size 13kb. The corresponding postcript file has gzipped size 70kb. Submitted from: mmartins@ugr.es The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.FA/0503076 or http://arXiv.org/abs/math.FA/0503076 or by email in unzipped form by transmitting an empty message with subject line uget 0503076 or in gzipped form by using subject line get 0503076 to: math@arXiv.org.