A \v{Z}ivaljevi\'c-Vre\'cica-Dolnikov-type theorem for super-Rado depth
Alexander Magazinov · arXiv (Cornell University) · 2016
A celebrated theorem proved by \v{Z}ivaljevi\'c and Vre\'cica, and, independently, by Dolnikov asserts that for $m$ measures in $\mathbb R^{n + m - 1}$ there exists a projection onto an $n$-space such that one can found a single point at depth at least $\tfrac{1}{d + 1}$ (the Rado bound) for all $n$-dimensional marginals. We consider a similar problem of projecting $m$ measures onto an $n$-subspace, but we request a greater depth of a point. It appears that for depth $\tfrac{1}{n + 1} + \tfrac{1}{3(n + 1)^3}$ the dimension threshold for the ambient space is polynomial in $m$ and $n$.