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\chapter{Levy's family and observable diameters}

In this section, we will explore how the results from Hayden's concentration of measure theorem can be understood in terms of observable diameters from Gromov's perspective and what properties it reveals for entropy functions.

We will try to use the results from previous sections to estimate the observable diameter for complex projective spaces.

\section{Observable diameters}

Recall from previous sections, an arbitrary 1-Lipschitz function $f:S^n\to \mathbb{R}$ concentrates near a single value $a_0\in \mathbb{R}$ as strongly as the distance function does.

\begin{defn}
  \label{defn:mm-space}

  Let $X$ be a topological space with the following:

  \begin{enumerate}
    \item $X$ is a complete (every Cauchy sequence converges)
    \item $X$ is a metric space with metric $d_X$
    \item $X$ has a Borel probability measure $\mu_X$
  \end{enumerate}

  Then $(X,d_X,\mu_X)$ is a \textbf{metric measure space}.
\end{defn}

\begin{defn}
  \label{defn:diameter}

  Let $(X,d_X)$ be a metric space. The \textbf{diameter} of a set $A\subset X$ is defined as
  $$
  \diam(A)=\sup_{x,y\in A}d_X(x,y).
  $$
\end{defn}

\begin{defn}
  \label{defn:partial-diameter}

  Let $(X,d_X,\mu_X)$ be a metric measure space, For any real number $\alpha\leq 1$, the \textbf{partial diameter} of $X$ is defined as
  $$
  \diam(A;\alpha)=\inf_{A\subseteq X|\mu_X(A)\geq \alpha}\diam(A).
  $$
\end{defn}

This definition generalize the relation between the measure and metric in the metric-measure space. Intuitively, the space with smaller partial diameter can take more volume with the same diameter constrains.

However, in higher dimensions, the volume may tend to concentrates more around a small neighborhood of the set, as we see in previous chapters with high dimensional sphere as example. We can safely cut $\kappa>0$ volume to significantly reduce the diameter, this yields better measure for concentration for shapes in spaces with high dimension.


\begin{defn}
  \label{defn:observable-diameter}
  Let $X$ be a metric-measure space, $Y$ be a metric space, and $f:X\to Y$ be a 1-Lipschitz function. Then $f_*\mu_X=\mu_Y$ is a push forward measure on $Y$.
  
  For any real number $\kappa>0$, the \textbf{$\kappa$-observable diameter with screen $Y$} is defined as

  $$
  \obdiam_Y(X;\kappa)=\sup\{\diam(f_*\mu_X;1-\kappa)\}
  $$

  And the \textbf{obbservable diameter with screen $Y$} is defined as

  $$
  \obdiam_Y(X)=\inf_{\kappa>0}\max\{\obdiam_Y(X;\kappa)\}
  $$

  If $Y=\R$, we call it the \textbf{observable diameter}.

\end{defn}

If we collapse it naively via
$$
  \inf_{\kappa>0}\obdiam_Y(X;\kappa),
$$
we typically get something degenerate: as $\kappa\to 1$, the condition ``mass $\ge 1-\kappa$'' becomes almost empty space, so $\diam(\nu;1-\kappa)$ can be forced to be $0$ (take a tiny set of positive mass), and hence the infimum tends to $0$ for essentially any non-atomic space.

This is why one either:
\begin{enumerate}
  \item keeps $\obdiam_Y(X;\kappa)$ as a \emph{function of $\kappa$} (picking $\kappa$ to be small but not $0$), or
  \item if one insists on a single number, balances ``spread'' against ``exceptional mass'' by defining $\obdiam_Y(X)=\inf_{\kappa>0}\max\{\obdiam_Y(X;\kappa),\kappa\}$ as above.
\end{enumerate}
The point of the $\max\{\cdot,\kappa\}$ is that it prevents cheating by taking $\kappa$ close to $1$: if $\kappa$ is large then the maximum is large regardless of how small $\obdiam_Y(X;\kappa)$ is, so the infimum is forced to occur where the exceptional mass and the observable spread are small.

Few additional proposition in \cite{shioya2014metricmeasuregeometry} will help us to estimate the observable diameter for complex projective spaces.

\begin{prop}
  \label{prop:observable-diameter-domination}
  Let $X$ and $Y$ be two metric-measure spaces and $\kappa>0$, and let $f:Y\to X$ be a 1-Lipschitz function ($Y$ dominates $X$, denoted as $X\prec Y$) then:

  \begin{enumerate}
    \item
    $
    \diam(X,1-\kappa)\leq \diam(Y,1-\kappa)
    $
    \item $\obdiam(X;-\kappa)\leq \diam(X;1-\kappa)$, and $\obdiam(X)$ is finite.
    \item
    $
    \obdiam(X;-\kappa)\leq \obdiam(Y;-\kappa)
    $
  \end{enumerate}
\end{prop}

\begin{proof}
  Since $f$ is 1-Lipschitz, we have $f_*\mu Y=\mu_X$. Let $A$ be any Borel set of $Y$ with $\mu_Y(A)\geq 1-\kappa$ and $\overline{f(A)}$ be the closure of $f(A)$ in $X$. We have $\mu_X(\overline{f(A)})=\mu_Y(f^{-1}(\overline{f(A)}))\geq \mu_Y(A)\geq 1-\kappa$ and by the 1-lipschitz property, $\diam(\overline{f(A)})\leq \diam(A)$, so $\diam(X;1-\kappa)\leq \diam(A)\leq \diam(Y;1-\kappa)$.

  Let $g:X\to \R$ be any 1-lipschitz function, since $(\R,|\cdot|,g_*\mu_X)$ is dominated by $X$, $\diam(\R;1-\kappa)\leq \diam(X;1-\kappa)$. Therefore, $\obdiam(X;-\kappa)\leq \diam(X;1-\kappa)$.
 
  and
  $$
  \diam(g_*\mu_X;1-\kappa)\leq \diam((f\circ g)_*\mu_Y;1-\kappa)\leq \obdiam(Y;1-\kappa)
  $$
\end{proof}

\begin{prop}
  \label{prop:observable-diameter-scale}
  Let $X$ be an metric-measure space. Then for any real number $t>0$, we have
  
  $$
  \obdiam(tX;-\kappa)=t\obdiam(X;-\kappa)
  $$

  Where $tX=(X,tdX,\mu X)$.
\end{prop}

\begin{proof}
  $$
  \begin{aligned}
    \obdiam(tX;-\kappa)&=\sup\{\diam(f_*\mu_X;1-\kappa)|f:tX\to \R \text{ is 1-Lipschitz}\}\\
    &=\sup\{\diam(f_*\mu_X;1-\kappa)|t^{-1}f:X\to \R \text{ is 1-Lipschitz}\}\\
    &=\sup\{\diam((tg)_*\mu_X;1-\kappa)|g:X\to \R \text{ is 1-Lipschitz}\}\\
    &=t\sup\{\diam(g_*\mu_X;1-\kappa)|g:X\to \R \text{ is 1-Lipschitz}\}\\
    &=t\obdiam(X;-\kappa)
  \end{aligned}
  $$
\end{proof}

\subsection{Observable diameter for class of spheres}

In this section, we will try to use the results from previous sections to estimate the observable diameter for class of spheres.

\begin{theorem}
  \label{thm:observable-diameter-sphere}
  For any real number $\kappa$ with $0<\kappa<1$, we have
  $$
  \obdiam(S^n(1);-\kappa)=O(\sqrt{n})
  $$
\end{theorem}

\begin{proof}
  First, recall that by maxwell boltzmann distribution, we have that for any $n>0$, let $I(r)$ denote the measure of standard gaussian measure on the interval $[0,r]$. Then we have
  
  $$
  \begin{aligned}
    \lim_{n\to \infty} \obdiam(S^n(\sqrt{n});-\kappa)&=\lim_{n\to \infty} \sup\{\diam((\pi_{n,k})_*\sigma^n;1-\kappa)|\pi_{n,k} \text{ is 1-Lipschitz}\}\\
    &=\lim_{n\to \infty} \sup\{\diam(\gamma^1;1-\kappa)|\gamma^1 \text{ is the standard gaussian measure}\}\\
    &=\diam(\gamma^1;1-\kappa)\\
    &=2I^{-1}(\frac{1-\kappa}{2})\text { cutting the extremum for normal distribution}\\
  \end{aligned}
  $$

  By proposition \ref{prop:observable-diameter-scale}, we have

  $$
  \obdiam(S^n(\sqrt{n});-\kappa)=\sqrt{n}\obdiam(S^n(1);-\kappa)
  $$

  So $\obdiam(S^n(1);-\kappa)=\sqrt{n}(2I^{-1}(\frac{1-\kappa}{2}))=O(\sqrt{n})$.
\end{proof}

From the previous discussion, we see that the only remaining for finding observable diameter of $\C P^n$ is to find the lipchitz function that is isometric with consistent push-forward measure.

To find such metric, we need some additional results from previous sections.

A natural measure for $\C P^n$ is the normalized volume measure on $\C P^n$ induced from the Fubini-Study metric. \cite{lee_introduction_2018} Example 2.30

\begin{defn}
  \label{defn:fubini-study-metric}

  Let $n$ be a positive integer, and consider the complex projective space $\C P^n$ defined as the quotient space of $\C^{n+1}$ by the equivalence relation $z\sim z'$ if there exists $\lambda \in \C$ such that $z=\lambda z'$. The map $\pi:\C^{n+1}\setminus\{0\}\to \C P^n$ sending each point in $\C^{n+1}\setminus\{0\}$ to its span is surjective smooth submersion.

  Identifying $\C^{n+1}$ with $\R^{2n+2}$ with its Euclidean metric, we can view the unit sphere $S^{2n+1}$ with its roud metric $\mathring{g}$ as an embedded Riemannian submanifold of $\C^{n+1}\setminus\{0\}$. Let $p:S^{2n+1}\to \C P^n$ denote the restriction of the map $\pi$. Then $p$ is smooth, and its is surjective, because every 1-dimensional complex subspace contains elements of unit norm.

\end{defn}

There are many additional properties for such construction, we will check them just for curiosity.

We need to show that it is a submersion.

\begin{proof}
   Let $z_0\in S^{2n+1}$ and set $\zeta_0=p(z_0)\in \C P^n$. Since $\pi$ is a smooth submersion, it has a smooth local section $\sigma: U\to \C^{n+1}$ defined on a neighborhood $U$ of $\zeta_0$ and satisfying $\sigma(\zeta_0)=z_0$ by the local section theorem (Theorem \ref{theorem:local_section_theorem}). Let $v:\C^{n+1}\setminus\{0\}\to S^{2n+1}$ be the radial projection on to the sphere:

   $$
   v(z)=\frac{z}{|z|}
   $$

   Since dividing an element of $\C^{n+1}$ by a nonzero scalar does not change its span, it follows that $p\circ v=\pi$. Therefore, if we set $\tilde{\sigma}=v\circ \sigma$, then $\tilde{\sigma}$ is a smooth local section of $p$. Apply the local section theorem (Theorem \ref{theorem:local_section_theorem}) again, this shows that $p$ is a submersion.

   Define an action of $S^1$ on $S^{2n+1}$ by complex multiplication:

   $$
   \lambda (z^1,z^2,\ldots,z^{n+1})=(\lambda z^1,\lambda z^2,\ldots,\lambda z^{n+1})
   $$

   for $\lambda\in S^1$ (viewed as complex number of norm 1) and $z=(z^1,z^2,\ldots,z^{n+1})\in S^{2n+1}$. This is easily seen to be isometric, vertical, and transitive on fibers of $p$.

   By (Theorem \ref{theorem:riemannian-submersion}). Therefore, there is a unique metric on $\C P^n$ such that the map $p:S^{2n+1}\to \C P^n$ is a Riemannian submersion. This metric is called the Fubini-study metric.
\end{proof}

\subsection{Observable diameter for complex projective spaces}

Using the projection map and Hopf's fibration, we can estimate the observable diameter for complex projective spaces from the observable diameter of spheres.

\begin{theorem}
  \label{thm:observable-diameter-complex-projective-space}
  For any real number $\kappa$ with $0<\kappa<1$, we have
  $$
  \obdiam(\mathbb{C}P^n(1);-\kappa)\leq O(\sqrt{n})
  $$
\end{theorem}

\begin{proof}
  Recall from Example 2.30 in \cite{lee_introduction_2018}, the Hopf fibration $f_n:S^{2n+1}(1)\to \C P^n$ is 1-Lipschitz continuous with respect to the Fubini-Study metric on $\C P^n$. and the push-forward $(f_n)_*\sigma^{2n+1}$ coincides with the normalized volume measure on $\C P^n$ induced from the Fubini-Study metric.

  By proposition \ref{prop:observable-diameter-domination}, we have $\obdiam(\mathbb{C}P^n(1);-\kappa)\leq \obdiam(S^{2n+1}(1);-\kappa)\leq O(\sqrt{n})$.

\end{proof}

\section{Use entropy function as estimator of observable diameter for complex projective spaces}

In this section, we wish to use observable diameter to estimate the statics of thermal dynamics of some classical systems.



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