  We determine the maximal hyperplane sections of the regular $n$-simplex, if
the distance of the hyperplane to the centroid is fairly large, i.e. larger
than the distance of the centroid to the midpoint of edges. Similar results for
the n-cube and the l_1-ball were obtained by Moody, Stone, Zach and Zvavitch
and by Liu and Tkocz. The maximal hyperplanes in these three cases are
perpendicular to the vectors from the centroid to the vertices. For smaller
distances -- in a well-defined range -- we show that these hyperplane sections
are at least locally maximal. We also determine the hyperplane sections of the
simplex, the cross-polytope and the cube which have maximal perimeter, i.e.
maximal volume intersection with the boundary of the convex body.
