It won't be popular. Why? Because it will be harder to clean on the outside.

Interesting that they don't show in inside of the pans. If those fins are pressed into the sides so that you get the reverse pattern inside I can foresee a lot of effort being needed to keep the inside clean.
As it happens I have a set of pans from the 50s that have a truncated conical shape that also heat the contents faster than parallel sided pans just as the blurb said (if I remember the blurb that came with them correctly) so this general idea isn't new.

Seems to me their one advantage only happens with gas stoves. A lot of consumers have electric stoves, which wouldn't benefit much from the fins.
In my estimation, the most efficient stove and cookware combination would be induction cooktop under special pots and pans that are multi-ply stainless with aluminum core, plus a seasonable carbon steel inner surface for saute and other fry pans. The sides and lid of the cookware should be insulated, perhaps with glass fiber batting, to vastly reduce heat loss to the ambient air. My induction cooker already heats stainless and carbon steel cookware at least as fast as a gas burner would. With insulated sides, it would be unbeatable.

@ Ivan,
"If those fins are pressed into the sides so that you get the reverse pattern inside I can foresee a lot of effort being needed to keep the inside clean."
They are not.

love it. I imagine the inside would be smooth. But i'd double-check before purchasing.

It certainly gives more area to absorb heat from the fire. I don't see it providing an improvement on non flame heat sources unless the inside is a cleaning nightmare.

If you don't have a gas (or open fire) cooking system it seems to be a waste of money. An outer flared 'halo' base ring - detachable for cleaning - would channel the heat just as well, and probably easier to make.

I'm just wondering if they are suitable for use on an induction cooktop and, if so, whether they have found any improvements of this design over the normal pans. Personally, I like the way Gadgeteer thinks. Why haven't pans been designed with integrated insulation?

In a sane society we would slowly phase out old pans and such with new more efficient designs and save mega tons of gas. Instead this will be only at max in a small amount of pans as a niche market or something.

Alternatively, make the bottom of the pot concave, ie bowed upwards in the center. The hot air would concentrate in the center pocket underneath the pot.
Or an outer skirt, a complement of above, as mentioned by The Skud
Of course both solutions only really suitable for gas.
Realistically an insulated outer cradle (nest) like a slow cooker or rice cooker using electric coil or gas would be most efficient. But a bit of a pain to clean unless the insulated outer is completely concealed in a smooth washable tub.