In 2009, Camillo De Lellis and László Székelyhidi Jr. proved a non-uniqueness result for weak solutions to the incompressible Euler equation, through a convex integration method coming from geometry (Nash 54', Gromov 86'). A consequence of their proof is the following surprising result : the set of weak solutions to the equation is dense in $L^{\infty}_t L^2_x$ for the weak topology of this space. By generalizing the geometric frame of their approach, I will explain how to show that the set of weak solutions is path-connected for the strong topology, and I will give the main ideas of the proof of De Lellis and Székelyhidi.