A Measure Concentration Effect for Matrices of High, Higher, and Even Higher Dimension

Let $n>m$, and let $A$ be an $(m\times n)$-matrix of full rank. Then obviously the estimate $\|Ax\|\leq\|A\|\|x\|$ holds for euclidean norm $x$ $Ax$ spectral as assigned matrix norm. We study sets all which, fixed $\delta<1$, conversely $\|Ax\|\geq\delta\,\|A\|\|x\|$ holds. It turns out that these fill, in high-dimensional case, almost complete space once $\delta$ falls below a bound depends on extremal singular values ratio dimensions. This effect has much to do with random projection theorem, which plays important role data sciences. As byproduct, we calculate probabilities this theorem deals exactly.