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In manual is see:

Polygons inherit a radius from b2Shape. The radius creates a skin around the polygon. The skin is used in stacking scenarios to keep polygons slightly separated. This allows continuous collision to work against the core polygon. The polygon skin helps prevent tunneling by keeping the polygons separated.

From this I don't understand what problem is being solved with the skin? What if there were not skins, what would happen?

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Polygon skin is used in Continuous Collision Detection (CCD) and is something internal to Box2D.

As says Erin Catto:

Polygon skin is a buffer for continuous collision detection. CCD breaks down when shapes overlap. Need a small separation. You shouldn't need to tune it if you use meters.

Here is more information how CCD works in Box2D.

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