Well, where I live we don't have any Apple Geniuses (Hong Kong), so eventually I went to a local authorized dealer from whom I've bought other stuff in the past and am known. They graciously hooked up the cable to a television in their shop. When it didn't work there either, they suggested that I take it to the local (Hong Kong) Apple authorized repair place. The folk at the repair place took the cable and said that they'd give me a replacement. Unfortunately, they didn't have any in stock, so I'm waiting for notification to pick up the new cable.
Luckily I had purchased the cable from the Apple online store, so I could log in and show the purchase details. I hadn't kept the box or receipts, because I never dreamt one could have a problem with something as simple as a cable. Once they saw the purchase, the didn't ask any other questions.
I did have some concerns that they might ask questions about scratches on the connectors. Frankly, while the connectors are attractive, the smooth metal cylinders are not very functional. Once you firmly seat the connectors, they are so smooth that they're hard to grasp with your fingers. With this problem, I had shifted them from one set of inputs (input #1) to another (input#2) and so forth to rule out problems with any particular input. I ended up resorting to use of pliers to pull out the connectors -- better that than pull on the wires and eventually pull them loose from the jacks. Here is one place where Apple's emphasis on good looking design made the product less usable (not to mention the defect).
I still can't see how a cable could be defective, but you live and learn. I'd have to suggest that you return your cable, as well.