Section of ellipsoids

Question

I am reading some survey about euclidean sections of convex bodies(http://arxiv.org/abs/1110.6401). It is written that any $k$-dimensional ellipsoid easily seen to have a $k/2$-dimensional section which is a multiple of an Euclidean ball. Can anyone tell me the reference of this fact or how to prove it?

Answer 1

This is a linear algebra question: the claim is equivalent to "given a $k \times k$ positive definite matrix $A$, there is a rank $k/2$ orthogonal projection $P$ such that $PAP$ is a multiple of $P$".

Proof: we show more precisely that, if $\lambda>0$ is a median of the set of eigenvalues of $A$ (i.e. $A$ has at least $k/2$ eigenvalues $\leq \lambda$, and at least $k/2$ eigenvalues $\geq \lambda$, counting multiplicity), then there is rank $k/2$ orthogonal projection such that $PAP = \lambda P$.

The case $k=2$ is easy. The general case reduces to it: we can assume that $P$ is diagonal, with first $k/2$ diagonal entries $\geq \lambda$ and last $k/2$ entries $\leq \lambda$. Apply the case $k=2$ to the restriction of $A$ to the space spanned by the $i$-th and $k+1-i$-th vectors of the canonical basis.