  The notion of the H\"older convolution is introduced. The main result is
that, under general conditions on functions L_1, ..., L_n, the function inverse
to the Legendre--Fenchel transform of the H\"older convolution of L_1, ..., L_n
coincides with the sum of the inverses of the Legendre--Fenchel transforms of
the individual functions L_1, ..., L_n. Applications to probability theory are
presented. In particular, an upper bound on the quantiles of the distribution
of the sum of random variables is given.
