You could slope the gutters (running East/West) all to the East side (or West, if that's more suitable, I'll stick with East for this answer), then at the end of the house, instead of ending in a pipe going straight down the corner, you would use a couple of elbows to run it past the East side wall into a piece of downspout running at a gentle slope across the East wall to the center. At the center, it would meet up with a downspout coming from the other eave end, where the two downspouts would dump into a collector then run down a single downspout to the ground and disperse water to wherever you want to direct it.
What you might find, however, is that your entire roof collects far too much water for the current gutter/downspout sizes to handle, so you'd have to go with larger gutters (currently, I'll bet there's a high point in the center of each North/South eave so that roughly half the water runs to each corner) to handle all the water coming off of each roof plane. Then you'd have to have larger downspouts to handle that side's water running at a slope across the East gable end wall. Then you'd have to have an even larger downspout to handle all the water landing on the entire roof coming through one single downspout.
You might also find that your fascia boards (that's the horizontal trim board at the end of the eave) might not be tall enough to support the entire (1/8" per foot) slope necessary to get good drainage from the gutter on each eave. As a quick idea, if your North eave is 32' long, you'll need to have a drop of 4" from West to East to drain to that single downspout elbow to make it around the corner to the East wall. If you have a downspout at each corner, you only need a drop of 2" (starting at the center, lowering toward each corner) to drain into 2 downspouts.
Are there options for doing this with a single downspout? Yes.
Does it make sense? Not to me, but that's opinion based and technically that's off-topic at all StackExchange sites.