Let $n\in\mathbb N^*$. We consider $\mathbb R^n$ with its euclidean structure. Let's begin with some notations :
- $C_n = [-1/2,1/2]^n$ is the unit cube in $\mathbb R^n$
- $\mathbb S^{n-1}$ denotes the unit sphere in $\mathbb R^n$.
- For $\xi \in \mathbb S^{n-1}$ we denote $H(\xi)$ the $(n-1)$-hyperplane $H(\xi) = \{\xi\}^{\bot}$
- $\Pi(\xi)$ denotes the orthegonal projection of $C_n$ onto the hyperplane $H(\xi)$.
- $v_{n-1}(X)$ denotes the $(n-1)$ volume of $X\subset \mathbb R^n$ a $(n-1)$-dimensionnal set.
So here's the problem:
We consider a parallelotope $P$ (i.e. $P = M(C_n)$ for $M\in \mathrm{GL}_n(\mathbb R)$). What is the maximum of $v_{n-1}(\Pi(\xi))$ and wich is the $\xi$ that give this maximum?
What I did so far:
Nothing successful
Difficulties encountered:
- How to extent the result for $C_n$ to parallelotopes?
EDIT: thanks to a comment, I know that an assumption that I made was false.