Sections of simplices and cylinders - Volume formulas and estimates
We investigate sections of simplices and generalized cylinders. We are interested in the volume of sections of these bodies with affine subspaces and give formulas and estimates for these volumes. For the regular n-simplex we state a general formula to compute the volume of the intersection with some k-dimensional subspace. A formula for central hyperplane sections was given by S. Webb. He also showed that the hyperplane through the centroid containing n-1 vertices gives the maximal volume. We generalize the formula to arbitrary dimensional sections that do not necessarily have to contain the centroid. And we show that, for a prescribed small distance of a hyperplane to the centroid, still the hyperplane containing n-1 vertices is volume maximizing. The proof also yields a new and short argument for Webb's result. The minimal hyperplane section is conjectured to be the one parallel to a face. We show that this hyperplane section is indeed minimal for dimensions n=2,3,4 and that it is a local minimum in general. Using results by Brehm e.a. we compute the average hyperplane section volume. For k-dimensional sections we give an upper bound. Finally we modify our volume formula to compute the section volume of irregular simplices. As an application we show that in odd dimensions larger than 4 there exist irregular simplices whose maximal section is not a face. A generalized cylinder is the Cartesian product of a n-dimensional cube and a m-dimensional ball of radius r. We study the behavior of the hyperplane section volume depending on the radius of the cylinder. First we show for the 3-dimensional cylinder that always a truncated ellipse gives the maximal volume. This is done by elementary geometric considerations and calculus. For the generalized cylinder we use the Fourier transform to derive an explicit formula. Then we estimate this by Hölder's inequality. Finally it remains to prove an integral inequality that is similar to the inequality of K. Ball for the cube.