Let $X$ be a finite dimensional vector space and $ C $ be a orgin symmetric convex body. I would like to describe the notion that, given any basis $ (e_k)_{k=1}^{n} $ of $X$ and any vector $ x = \sum_{k=1}^{n} x_{k} e_k \in B $, and any $ y = \sum_{k=1}^{n} y_k e_k \in \partial B$ in the boundary of $B$, there exists a sequence $ (m_{k})_{k=1}^{n} $ such that for all $ 1 \leq k \leq n $, $ |x_k| \leq | m_k| \leq |y_k| $ holds. I am kind of sure this statement is true but however not sure if there is any theorem I could quote or how to prove this claim formally as I am new to convex analysis. Could anyone provide me with some clues?
I also suspect that the claim does not hold for general origin symmetric convex body, would this be correct as well?
Thanks in advance!
Edit: It is sufficient that the statement, given any vector $ x = \sum_{k=1}^{n} x_k e_k \in B $, and all $ y = \sum_{k=1}^{n} y_k e_k \in \partial B $ we have $ |x_k| \leq |y_k|$ for all $ 1 \leq k \leq n $. The sequence $ (m_k)_{k=1}^{n} $ needs not to be there.
Edit: As mentioned by the comments the claim is false. What if I adjust the claim to at least one coordinate? That is, for some $ 1 \leq k \leq n $ we must have $ |x_k| \leq |y_k| $?
Not even one coordinate needs to be majorized in this way. Here is a rhombus in which $(4, 4)$ is an interior point and $(2, 2/5)$ is a boundary point.