We define two metrics $d_{1,\alpha }$ and $d_{\infty ,\alpha }$ on each Schreier family ${\mathcal{S}}_{\alpha }$, $\alpha <{\omega }_1$, with which we prove the following metric characterization of the reflexivity of a Banach space $X$: $X$ is reflexive if and only if there is an $\alpha <{\omega }_1$ such that there is no mapping $\Phi :{\mathcal{S}}_{\alpha }\to X$ for which $$c{d}_{\infty ,\alpha }(A,B)\le \Vert \Phi \left(A\right)-\Phi \left(B\right)\Vert \le C{d}_{1,\alpha }(A,B)\text{for all}A,B\in {\mathcal{S}}_{\alpha }.$$ Additionally we prove, for separable and reflexive Banach spaces $X$ and certain countable ordinals $\alpha$, that $max\hspace{0.17em}(Sz(X),Sz(X^{\ast }\left)\right)\le \alpha$ if and only if $({\mathcal{S}}_{\alpha },d_{1,\alpha })$ does not bi-Lipschitzly embed into $X$. Here $Sz\left(Y\right)$ denotes the Szlenk index of a Banach space $Y$.

Let $E\subseteq {\mathbb{P}}^2$ be a complex rational cuspidal curve contained in the projective plane. The Coolidge–Nagata conjecture asserts that $E$ is Cremona-equivalent to a line, that is, it is mapped onto a line by some birational transformation of ${\mathbb{P}}^2$. The second author recently analyzed the log minimal model program run for the pair $(X,\frac{1}{2}D)$, where $(X,D)\to ({\mathbb{P}}^2,E)$ is a minimal resolution of singularities, and as a corollary he proved the conjecture in the case when more than one irreducible curve in ${\mathbb{P}}^2\setminus E$ is contracted by the process of minimalization. We prove the conjecture in the remaining cases.

This is a continuation of an earlier work in which we proposed a problem of minimizing normalized volumes over $\mathbb{Q}$-Gorenstein Kawamata log terminal singularities. Here we consider its relation with K-semistability, which is an important concept in the study of Kähler–Einstein metrics on Fano varieties. In particular, we prove that for a $\mathbb{Q}$-Fano variety $V$, the K-semistability of $(V,-K_V)$ is equivalent to the condition that the normalized volume is minimized at the canonical valuation ${\mathrm{ord}}_V$ among all ${\mathbb{C}}^{\ast }$-invariant valuations on the cone associated to any positive Cartier multiple of $-K_V$. In this case, we show that ${\mathrm{ord}}_V$ is the unique minimizer among all ${\mathbb{C}}^{\ast }$-invariant quasimonomial valuations. These results allow us to give characterizations of K-semistability by using equivariant volume minimization, and also by using inequalities involving divisorial valuations over $V$.