The present invention relates to a composition in the form of an oil-in-water emulsion, comprising, in a cosmetically acceptable medium: A) a continuous aqueous phase and B) an oily phase dispersed in the said aqueous phase and comprising at least one hydrocarbon-based oil; C) at least one mixture consisting of: i) at least one nonionic surfactant containing a saturated linear hydrocarbon-based chain comprising at least 16 carbon atoms, the said surfactant being other than a fatty alcohol, and ii) at least one fatty alcohol chosen from: —a pure fatty alcohol comprising more than 16 carbon atoms; —a mixture consisting exclusively of fatty alcohols containing at least 16 carbon atoms and comprising at least 50% by weight of fatty alcohol containing at least 18 atoms relative to the weight of the fatty alcohol mixture; and D) at least one wax with a melting point of greater than 45° C. comprising one or more C40-C70 ester compounds and not comprising any C20-C39 ester compounds, in an amount from 1 to 10% by weight relative to the total weight of the composition and E) at least one water-soluble polysaccharide; the said composition having a hardness measured at 32° C. and at a humidity of 40% ranging from 20 kPa to 200 kPa and preferably ranging from 25 kPa to 150 kPa. The present invention also relates to a cosmetic process for treating and/or caring for human keratin materials, characterized in that it consists in applying to the surface of the keratin material a composition as defined previously. The present invention also relates to a cosmetic process for treating human perspiration and/or perspiration-related body odor, which consists in applying to the surface of a human keratin material a composition as defined previously comprising at least one deodorant active agent and/or one antiperspirant active agent.