  Let $k\geq 2$ be an integer. In the spirit of Kolesnikov-Werner \cite{KW},
for each $j\in\{2,\ldots,k\}$, we conjecture a sharp Santal\'{o} type
inequality (we call it $j$-Santal\'{o} conjecture) for many sets (or more
generally for many functions), which we are able to confirm in some cases,
including the case $j=k$ and the unconditional case. Interestingly, the
extremals of this family of inequalities are tuples of the $l_j^n$-ball. Our
results also strengthen one of the main results in \cite{KW}, which corresponds
to the case $j=2$. All members of the family of our conjectured inequalities
can be interpreted as generalizations of the classical Blaschke-Santal\'{o}
inequality. Related, we discuss an analogue of a conjecture due to K. Ball
\cite{Ball-conjecture} in the multi-entry setting and establish a connection to
the $j$-Santal\'{o} conjecture.
