  In this article, we address the following question: Is it true that the
spatial numerical range (SNR) $V_A(a)$ of an element $a$ in a normed algebra
$(A, \|\cdot\|)$ is always convex? If the normed algebra is unital, then it is
convex \cite[Theorem 3, P.16]{BoDu:71}. In non-unital case, we believe that the
problem is still open and its answer seems to be negative. In search of such a
normed algebra, we have proved that the SNR $V_A(a)$ is convex in several
non-unital Banach algebras.
