  The main goal of this paper is to present a series of inequalities connecting
the surface area measure of a convex body and surface area measure of its
projections and sections. We present a solution of a question from S. Campi, P.
Gritzmann and P. Gronchi regarding the asymptotic behavior of the best constant
in a recently proposed reverse Loomis-Whitney inequality. Next we give a new
sufficient condition for the slicing problem to have an affirmative answer, in
terms of the least "outer volume ratio distance" from the class of intersection
bodies of projections of at least proportional dimension of convex bodies.
Finally, we show that certain geometric quantities such as the volume ratio and
minimal surface area (after a suitable normalization) are not necessarily close
to each other.
