What's the advantage of the F4U Corsair's gull wing design?
The wing shape in the F4U Corsair is called the inverted gull wing. The main reasons for use of this is the large propeller used in the aircraft.

The Corsair design was in response to the US Navy RFP (Request for Proposal) in 1938, which mandated the following things:


It should have maximum attainable speed
The stall speed should be 70 mph
Minimum range of 1000 miles
Bombs should be carried under the wing (this was supposed to be dropped on aircraft formations from above, which didn't work out)


In order to achieve this, the Vought engineers used the largest available engine, the Prat and Whitney R2800 Double Wasp radial engine (this was done as the Navy, in a change of heart, vowed to consider designs with liquid-cooled engines, in contradiction with a policy settled in 1927 that required air-cooled engines for shipboard aircraft). Also, the wing area of F4U was the largest naval aircraft of its time.

During design, the oil cooler and supercharger inlets were moved to wing leading edges. In order to extract power from the engine, a large (~4m dia) propeller was chosen. This large propeller caused problems with ground clearance, which necessitated an increase in the landing gear length.

The Vought team overcame this by choosing an inverted gull wing, which starts out with a strong anhedral, and then curved upwards to strong dihedral, with landing gear at the lowest point in the bend. 


"F4U-Corsair OE-EAS OTT 2013 04 main landing gear" by Julian Herzog. Licensed under CC BY 4.0 via Commons.

The main advantages of this configuration are,


Shortened landing gear
Elimination of the need for wing fillets, and reducing drag as the wing and fuselage were perpendicular.
Simplified wing folding (the wings were folded at the lowest point), with the folded wings nearly at the height of the propellers. This enabled simple automatic mechanism for folding/unfolding, unlike the F6F Sto-wing which was manual, as the hydraulic mechanism added too much weight.



Source: www.airport-data.com

The main disadvantages were:


The design added weight compared to straight wing.
(To some people) It was ugly


This design has been used in a number of other aircraft too, mostly to reduce the length of the landing gear and carry large external stores, the most famous being the Junkers Ju-87 Stuka. 


"Bundesarchiv Bild 101I-646-5184-26, Russland, Flugzeug Junkers Ju 87 edit 1". Licensed under CC BY-SA 3.0 de via Commons.

The inverted gull wings were also used in Fairey Gannet, an ASW aircraft of Royal Navy.


Source: www.maritimequest.com

This was also used in some twin engined aircraft like the Caproni Ca.331.


Source: flickr.com

Note that in some books, this design is called the gull wing instead of inverted gull wing.