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There are a couple of suspension fork manufacturers that have adopted a fork design where the lowers have a reversed arch. I can't see that they claim any particular performance advantages for it, although there are some people on mountain biking forums saying they have improved torsional stiffness. Why would that be the case?
Also, how do they compare in muddy conditions, more/less resistance to clogging?
Looks like nobody has taken the geometry into account yet...
In the image below I have drawn the situation with highly exaggerated proportions.
All not-lowest-end suspension forks (at least the ones I know of) have a rake that is created by shifting the axle in front of the lower tubes. Let's take a wheel of radius r (solid blue circle) and add some clearance for the tire (dashed blue circle). Above that, the upper edge of the lower tubes shall be of equal height with respect to the upper tube's axis.
Here we can already see, that the wheel passes the front side of the lowers at a "higher" position than its back side. As the arches (green rectangles) have to span from one lower tube to the other over the clearance radius, we can see that the arch in the front side (the right one in the image) has to reach higher (higher green rectangle) above the lower tube's body than the one on the back side.
Technically speaking this means, that to build an arch with the same stiffness, less material (i.e. less weight) is needed on a reversed arch or at the same weight, the reversed arch can be built more massive and therefore stiffer. Unfortunately I don't have any measurement values about how much influence that really has. And obviously there are (or at least were, cf. heltonbike's answer) other drawbacks to be taken into account.
I have a Manitou Black with reverse arch (an old model) and it seems to me that the posts get dirtier than if it was a front arch model. Specifically, the grime that comes "in the air" hits the posts directly (without the arch acting as a sort of protection), and the mud that comes with the tire ends up getting everything dirty anyway, sooner or later. Also, since the O-ring around the posts is tilted backwards due to steering angle, mud actually tends to form a deposit behind the post.
Incidentally, I have rim brakes attached to the brake posts, and due to the arch being behind, I am forced to use a brake booster, otherwise the blades of the fork flex A LOT while braking, with poor performance.
In the end, I agree with the ones that think it's more marketing than engineering. Besides, if you Google "dual arch suspension fork" there are already some models around, and I have seen at least one "middle arch" fork, though I can't remember the brand.
One of the few things I can think of is that if the fork fails, or is improperly adjusted, having the brace in the back means there is some possibility that the brace will slam into the downtube or back of the headtube rather than the just the upper and lower of the fork meeting violently. That possibility depends on brace design, height of the upper, frame geometry, etc. But, putting it in front eliminates it as a possibility.
From a purely aesthetic standpoint, the brace replaces the area that rim brakes used to occupy on a fork. To some people (me included) placing the brace in the back just looks wrong, making the front of the fork look naked.
Can find references to anything, however I recall reading its pretty much a marketing gimmick. The engineer in me thinks I am sure you could probably do a middle or 2 arches shock if you wanted...
As far as resistance to clogging in mud, I expect the only dimension that counts is the tire to arch and tire to post distances. Bigger is better.
In my opinion it wouldn't give any advantage....except possibly with some lower end disc brakes where the rotas may tend to get very hot with hard use may make you slightly less prone to burned shins and ancles...other then that just a gimmic
