This paper completes a fundamental construction in Alexandrov geometry.Previously we gave a new construction of metric spaces with curvature boundseither above or below, namely warped products with intrinsic metric space baseand fiber, and with possibly vanishing warping functions -- thereby extendingthe classical cone and suspension constructions from interval base to arbitrarybase, and furthermore encompassing gluing constructions. This paper proves theconverse, namely, all conditions of the theorems are necessary. Note that inthe 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 basicproperties of warped products of metric spaces with possibly vanishing warpingfunctions, including new properties..
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