Vacuum brake booster with reduced dead stroke and brake system incorporating such booster

A skirt (10) can slide axially relative to a pneumatic piston (16) over a defined travel in a first phase of a braking action. The skirt transmits a pneumatic boost force to the hydraulic piston of a master cylinder via its central portion. The skirt takes the pneumatic piston along with it via an annular plate (100) when the skirt has completed the defined travel.