The Maximum of the Volume of a Cevian Simplex and its Parts

It is well known that the area of the cevian triangle for an interior point M of a triangle $γ$ is at most 1/4 of the area of the triangle $γ$. This can be generalized for n-dimensional simplex A₁A₂… Aₙ₊₁ using the concept of barycentric coordinates. The method also helps with the solution of a new optimization problem for the volume of the simplex MN₁… Nₙ, where N₁,… ,Nₙ are the intersection points of the lines AᵢM with the opposite hyperplanes of the simplex A₁A₂… Aₙ₊₁. This generalizes the recently considered special cases $n=2$ and $n=3$.