  We consider a general conic mixed-binary set where each homogeneous conic
constraint involves an affine function of independent continuous variables and
an epigraph variable associated with a nonnegative function, $f_j$, of common
binary variables. Sets of this form naturally arise as substructures in a
number of applications including mean-risk optimization, chance-constrained
problems, portfolio optimization, lot-sizing and scheduling, fractional
programming, variants of the best subset selection problem, and
distributionally robust chance-constrained programs. When all of the functions
$f_j$'s are submodular, we give a convex hull description of this set that
relies on characterizing the epigraphs of $f_j$'s. Our result unifies and
generalizes an existing result in two important directions. First, it considers
\emph{multiple general convex cone} constraints instead of a single
second-order cone type constraint. Second, it takes \emph{arbitrary nonnegative
functions} instead of a specific submodular function obtained from the square
root of an affine function. We close by demonstrating the applicability of our
results in the context of a number of broad problem classes.
