  This paper completes a fundamental construction in Alexandrov geometry.
Previously we gave a new construction of metric spaces with curvature bounds
either above or below, namely warped products with intrinsic metric space base
and fiber, and with possibly vanishing warping functions -- thereby extending
the classical cone and suspension constructions from interval base to arbitrary
base, and furthermore encompassing gluing constructions. This paper proves the
converse, namely, all conditions of the theorems are necessary. Note that in
the cone construction, both the construction and its converse are widely used.
We also show that our theorems for curvature bounded above and below,
respectively, are dual. We give the first systematic development of basic
properties of warped products of metric spaces with possibly vanishing warping
functions, including new properties..
