[Grishnakh]

I don't know, but just looking at that photo, and comparing to a traditional piston engine with its piston rings, I can see a bunch of problems:

1) there's a bunch of irregular shapes here to seal: each "chamber" has shorter seals in two places (the vertices), and two longer seals on the sides. So 4 separate seals per chamber (two of which are re-used for adjacent chambers). In a piston engine, each chamber has usually 2 or 3 seals, with one being the main seal (top of the piston) and the others being either a backup or an oil ring (for scraping off most of the oil). A circle is a much simpler design.

2) 4 seals means 4 places where there's going to be gaps where gases can escape between chambers, causing extra emissions. In a piston engine, there's only 1 gap in each piston ring, and you can rotate the rings so the gaps are not aligned. The sealing on piston engines is extremely good.

3) The piston engine is pretty simple really, as far as sealing and lubrication: oil is either squirted (some engines) or splashed up on the backsides of the pistons and drawn up on the piston walls to lubricate the rings as the piston rises, and then the oil is almost 100% cleaned off by the rings as it travels on the downward stroke. This mechanism works so well that modern piston engines burn a truly negligible amount of oil. Oil burning has always been a problem on rotaries, and looking at that photo it's fairly obvious why.

Mazda (and others) has been working on rotaries with very smart engineers for literally decades, and they still haven't fixed these problems. I don't think they ever will; these problems seem pretty fundamental to the layout of a rotary engine. They had some real promise and advantages, but like many technologies, the small details made it impossible to beat incumbent technologies. Piston engines were bad too, ages ago, before they figured out a lot of small details like how to seal them well, how to optimize intake and exhaust valve design and timing, how to best design combustion chambers to maximize compression ratio and eliminate hot spots causing detonation, etc. But some problems just can't be overcome.

[sjburt]

The other challenge vs reciprocating engines is that a piston engine cylinder has two volumes; the area above the piston where combustion occurs, and the area below, where there isn't any combustion. So during the stroke, you can spray the cylinder walls with lubricant, the bottom of piston, etc. So the seals can run on a layer of lubricant, and the piston can be cooled by the lubricant. In a rotary, you don't have that. Both sides of each seal are filled with fuel-air mixture, so to lubricate, you have to spray oil into the mixture. It's the same issue as with two strokes. They have to burn oil, and that is really tough on emissions, and it's really difficult to get heat away from the seal (the only place it can go is to the rotor).

[Grishnakh]

That's an excellent point; I mentioned squirting oil before, but I overlooked that it serves another purpose in cooling the piston and the seals. Lots of modern engines now have oil squirters underneath the pistons to do this.

However, since, as you say, they have to burn oil to cool and lubricate the seals, that does make me wonder if this kind of engine wouldn't work much better with a lubricating fuel, such as diesel or kerosene (jet fuel), rather than gasoline. Kerosene is used in jet turbine engines largely because it is a lubricating fuel, so it serves the purpose of both fueling and oiling. The engine in TFA appears to be a rotary derivative, but they do say that it's designed to run on JP-8 jet fuel, not gasoline.