Exact asymptotic moduli and a Daugavet example

The space $X=J^*$ defined by \eqref{eq:segment-norm} is infinite
dimensional, separable, and reflexive, and has no equivalent AUC norm.
Its given norm satisfies
\[
 \widehat\delta_X(t)\ge\sqrt{1+t^2/12}-1\qquad(t>0).
\]
For every integer $k\ge0$, every embedding of $D_k$ into $X$ has
distortion at least $\sqrt{1+k/12}$.

Let $Y$ be either the finite-height forest dual $X=J^*$ or the
infinite-height coordinate predual $B_\infty$. For every finite
ancestral $H$, every $x,y\in Y$ with $x=P_Hx$, and every $R\ge0$
with $\|x+y\|,\|x-y\|\le R$,
\begin{equation}
 \|Q_Hy\|\le2\sqrt{R^2-\|x\|^2}.
\end{equation}

For every integer $k\ge0$, every distortion-$D$ embedding of
$\mathcal D_k$ into either $X$ or $B_\infty$ satisfies
\[
 D^2\ge1+k/4.
\]

Let $E$ be an infinite-dimensional closed real subspace of
$L^1(\Omega,\mathbb P)$, where $\mathbb P$ is a probability measure.
Suppose its closed unit ball is totally bounded for convergence in
measure and its inherited norm has the Daugavet property.  Then every
$x\in S_E$ and $t>0$ satisfy
\[
 H_E(x,t)=\max\{t/2,t-1\},\qquad
 D_E(x,t)=\max\{0,t-2\}.
\]
Consequently these are also the formulas for
$\widehat\delta_E(t)$ and $\overline\delta_E(t)$.  The space $E$ admits
no equivalent AUC norm.

Each of the four completed spaces $X_K$ is a separable infinite-dimensional
real Banach space. For $K\in\{K_{\rm s},K_{\rm p},K_{\rm a},K_{\rm g}\}$
and $0<t<1$,
\begin{equation}
 \widehat\delta_{X_K}(t)\ge\sqrt{1+t^2/4}-1.
\end{equation}
None of these spaces admits an equivalent AUC norm.

The real Banach space $X_{\mathrm e}$ defined by \eqref{fr:tests} and
\eqref{fr:norm} satisfies
\begin{equation}
 \widehat\delta_{\|\cdot\|_{\mathrm e}}(t)
 \ge\frac{\sqrt{1+t^2/576}-1}{4}\qquad(0<t<1).
 
\end{equation}
It admits no equivalent AUC norm.  More precisely, if
$\alpha\|\cdot\|_{\mathrm e}\le N\le\beta\|\cdot\|_{\mathrm e}$,
where $0<\alpha\le\beta<\infty$, then
$\overline\delta_N(\alpha/(4\beta))=0$.

If $x,y\in X$, $P_Ax=x$ and $P_Ay=0$ for a finite head $A$, then,
for every $c>0$,
\begin{equation}
 \max\{\|x+y\|,\|x-y\|\}
 \ge\frac{\|x\|+c\|y\|/2}{\sqrt{1+4c^2}}.
 
\end{equation}
In particular the left side is at least
$\sqrt{\|x\|^2+\|y\|^2/16}$.

For every multiplier $W$ as above, the Banach space $X_W$ is infinite
dimensional and has no equivalent AUC norm.
For the exponential and Gaussian-square laws, the inherited norms satisfy
\begin{align}
 \whdelta_{X_{\mathrm{exp}}}(t)
 &\ge \frac{7t}{30}c(10/t)>0 &&(t>0),\\
 \whdelta_{X_{\mathrm{sq}}}(t)
 &\ge\frac{\PP\bigl(|G|\ge4\sqrt2(2+\sqrt2)/t\bigr)}{8}>0
 &&(t>0),\\
 \whdelta_{X_{\mathrm{sq}}}(t)
 &\ge\frac{\PP\bigl(|G|\ge8\sqrt6/t\bigr)}{4}>0
 &&(t>0).
\end{align}

For $N=N_\Sigma$ on $X_\Sigma$, or $N=N_J$ on $X_J$, one has
\begin{equation}
 \widehat\delta_N(t)\ge t^3/128\qquad(0<t<1).
\end{equation}
For $X_0$, a finite initial set $H$, a vector $x$ supported in $H$, and
$z\in F_H\setminus\{0\}$, one has
\begin{equation}
 \frac{N_0(x+z)+N_0(x-z)}2\ge N_0(x)
 +\frac{N_0(z)^3}{8(2N_0(x)+N_0(z))^2}.
\end{equation}
Each of $X_\Sigma,X_0,X_J$ admits no equivalent AUC norm.

For every $\varepsilon>0$ there is $\eta(\varepsilon)\in(0,1)$ such
that, for $x,z_1,z_2,\ldots\in X_J$,
\[
 \left.
 \begin{gathered}
 N_J(x+z_j),N_J(x-z_j)\le1\quad(j\ge1),\\
 N_J(z_i-z_j)\ge\varepsilon\quad(i\ne j)
 \end{gathered}
 \right\}
 \quad\Longrightarrow\quad N_J(x)\le1-\eta(\varepsilon).
\]
For $0<\varepsilon\le2$, one can take
$\eta(\varepsilon)=\min\{1/2,\gamma(2,\varepsilon/16)\}$.

Let $x\in X_{\mathrm v}$ have finite total support.  In each component
where $x^{(h)}\ne0$, choose a finite ancestral set $A_h$ containing
that support.  Suppose $y\in X_{\mathrm v}$ vanishes on every such
$A_h$.  With
$S=\|x+y\|_{\mathrm v}^2+\|x-y\|_{\mathrm v}^2$, one has
\begin{equation}
 S-2\|x\|_{\mathrm v}^2
 \ge c_6 S^{-2}\|y\|_{\mathrm v}^6,
 \qquad c_6=2\exp(-(\mathrm e-1))4^{-6}.
\end{equation}
The right side is zero when $S=0$.