For 1 ≤ m ≤ n, the mth decomposable numerical range of an n × n complex matrix A is defined and denoted by W(A) = {det(X*AX) : X is an n × m complex matrix such that X* X = I}. In this note, we determine the conditions on an essentially Hermitian matrix A (i.e., A is normal with collinear eigenvalues) such that W(A) is (i) convex, or (ii) simply connected. For a general A, we obtain a sufficient for W(A) to be starshaped. These answer some questions raised by Bebiano and the second author on the geometrical properties of the decomposable numerical range. © 1994, Taylor & Francis Group, LLC. All rights reserved.