Let $T\subset \mathbb{R}^n$ be a fixed simplex, $H\subset \mathbb{R}^n$ be a variable affine hyperplane. Is it true that the maximal area (i.e. the $(n-1)$-dimensional volume) of $T\cap H$ is attained when $H$ contains a facet of $T$?



Is a facet always a maximal area section of a simplex?
No, there is a $5$-dimensional simplex with a hyperplane section which is larger than any of its facets, see [Walkup, A simplex with a large cross-section, Am. Math. Monthly, January 1968](https://www.jstor.org/stable/pdf/2315102.pdf).
The idea is to squeeze a regular 5-simplex along the common perpendicular of two opposite 2-dimensional faces. The volume of the mid-section between these faces does not change, but it turns out that all facets become smaller than this cross-section if the scaling coefficient is small enough.


For more information see also the [PhD thesis of Dirksen](https://d-nb.info/1082537128/34).


