Warped products admitting a curvature bound

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.