A Pokémon with Gluttony will use a "pinch berry" when its HP drops to 50%, rather than the usual 25%.
Pinch berries are: Figy, Wiki, Mago, Aguav, Iapapa, Liechi, Ganlon, Salac, Petaya, Apicot, Lansat, Starf, Micle, and Custap. (Their descriptions state that they work "in a pinch.")