Skeletal Mesh+Shadow Glitch in UE4

Many thanks for your detailed answer.
I’ve made some first tests with triangulation in blender and it’s still the same bad result.
I’m using blender 2.79 and the FBX 7.4 binary option. Maybe this is the culprit?

As for the Flatten function, I’m not sure if it will not distort other parts of the mesh like the armpits, so it would only move the problem to another part of the mesh.

Concerning repainting the weights: How can I repaint the weights, when blender is showing everything in a perfect state and smooth while it looks glitchy in UE.
What I’m not understanding is why the mesh in ue4 is different from the mesh in blender? Is it a UE specific issue?

In my opinion something bad happens during import to ue4 (rounding error of vertex weights / precision issue), because opening the same fbx file in blender after
export is showing no problems with the mesh (beyond some bone issues, which don’t affect the mesh directly if you don’t change anything)
Is it possible to look up single vertex weights in ue (as numbers, not colors)?