The Bottom Line: JVC RS1 vs Sharp XV-Z20000
Each projector has loads of strengths, and a few weaknesses. Overall, the JVC RS1 has to be the winner, but not because it produces a significantly better picture - I consider the image quality to be definitely comparable between these two. The JVC wins by virtue of a lower price point, brighter picture, better performance out of the box (without adjusting), and that it will be quieter, and easier to place in your room. The JVC should fit well into about any room, in that regard, but more to the point, it can sit or mount anywhere the Sharp can, while the Sharp is far more limited, overall.
The Sharp, on the other hand, has more inputs, far more controls for fine tuning the image, and a touch more styling. Perhaps the biggest reason, though, for some to choose the XV-Z20000 over the RS1, will be its slightly sharper image. The other "biggie" is if you are planning to go full anamorphic widescreen (2.35:1) for Cinemascope with no letterboxing at the top and bottom. The Sharp supports anamorphic lenses while the JVC does not, out of the box. Even if that is your plan (2.35:1), that doesn't eliminate the JVC, though, since you can use it with an anamorphic lens, too, but you need to spend a couple thousand for an outboard processor. If you do that, though, the price advantage of the JVC will be lost.
In our spring projector comparison, (before we received the Sharp for review), the JVC RS1, took the Best In Class Award for over $5000 1080p projectors. I have been tempted to share that Best In Class Award with the Sharp, since it rivals the RS1 in picture quality, the JVC just has too many other advantages, so hangs on to exclusive ownership of the award (for now).