An increasing homeomorphism \(h\) on the real line \(\mathbb{R}\) is said to be strongly symmetric if it can extended a quasiconformal of upper half plane \(\mathbb{U}\) onto itself whose Beltrami coefficient \(\mu\) induces vanishing Carleson measure \(|\mu(z)|^2/y\,dx\,dy\) \(\mathbb{U}\). We will deal with class homeomorphisms and its Teichmüller space, which we call VMO-Teichmüller space. I...

Let L be a family of n blue lines in the real projective plane. Suppose that R is a collection of m red lines, different from the blue lines, and it is known that every edge in the arrangement A(L) is crosses by a line in R. We show that m ≥ n−1 3.5 . Our result is more general and applies to pseudo-line arrangements A(L), and even weaker assumptions are required for R.