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z<&e^i+;2b_i@AbFBMt?!>+*9oixgZRBDb?cc;JLG+FvKq`(AiwwlKRm?Y+-n=7f-z z$hUj_Vn6!)&^bCCU}xA4@UD&FSiw??^k7CtuQbC=cM|~nmU)fQcl}p}G;>m~4Cf-M z>ku#BAtK8&0xy&?-r|7Iz2lLbgW?yNqRfS!kktabV{C0aB7+ipx|&jR zZOjNa@T-2m(K!4=-)uh{`cIP79{2YnyGS1Hx;~V6U!mx3p%GPf&WEi_UPo7RkBuaG zYu}hjvi}Z?HCOkkL6_6i=8)5yaaJ%-IY$)ZmHLQSt^md=jZ(`mj?d^sKH_fFeEYjz z!uZrlQ3R` z>(aTJWFo-Q97A{LTVrcu(t3$+GFsu?<^Csw(JHoAEIbXedHdp3zNk6{`Z!cl#_@kE z+hXZI$y)anui*GhlTUsERR45X=t&i@wCB$4q2_m@F!PJG#u5(QHs-H=p4AXBj;IF{bAjXE!;B&vT zJA{czxrRGKhcCgS32O=0{ZJ_$XchjP{7C$2p{FU5Gwg1rIV+Lxk3qjyfN{>47n<+e zqx@)VT=tkC|nr^asb3nw|%(v)F8gZGv?>yJ_;HtWb;x z{BaJw>4Dx=d(j&D-SCFzIYNKm{VJMy(R1V?@Lh3BoE{weLp4C}$_E&?=J%i8?1@;s zH)P<%)4dj3%Yi(}a&_-R*pKa$5A?flgZ&6Gr%b1WERJbO*x~i=CQiT6 um=ibFqrrCgsWmo=#l;AVE-p7Ec9s=b$P=NX0dlZm}h>r1f(EkA+0A0$, +\begin{frame}{Observable diameter: the inner definition} + \begin{block}{Partial diameter on $\mathbb{R}$} + Let $\nu$ be a Borel probability measure on $\mathbb{R}$ and let $\alpha \in (0,1]$. + The \textbf{partial diameter} of $\nu$ at mass level $\alpha$ is $$ - \obdiam_{\mathbb{R}}(X;-\kappa) - = - \sup_{f\in \operatorname{Lip}_1(X,\mathbb{R})} - \diameter(f_*\mu_X;1-\kappa). + \diameter(\nu;\alpha):= + \{\diameter(A):A \subseteq \mathcal{B}(\mathbb{R}), + \nu(A)\ge \alpha + \}, + $$ + where + $$ + \diameter(A):=\sup_{x,y\in A}|x-y|. $$ \end{block} + \vspace{0.4em} \begin{itemize} - \item It asks how concentrated every $1$-Lipschitz real observable must be. - \item In the thesis, entropy is used as a concrete observable-diameter proxy. - \item Hopf fibration lets us compare $\mathbb{C}P^n$ with spheres. + \item This asks for the shortest interval-like region containing at least $\alpha$ of the total mass. + \item So $\diameter(\nu;1-\kappa)$ measures how tightly we can capture \emph{most} of the distribution, allowing us to discard a set of mass at most $\kappa$. \end{itemize} + +\end{frame} + +\begin{frame}{Observable diameter of a metric-measure space} + \begin{block}{Definition} + Let $X=(X,d_X,\mu_X)$ be a metric-measure space and let $\kappa>0$. + The \textbf{observable diameter} of $X$ is + $$ + \obdiam_{\mathbb{R}}(X;-\kappa) + := + \sup_{f\in \operatorname{Lip}_1(X,\mathbb{R})} + \diameter(f_*\mu_X;1-\kappa), + $$ + where $\operatorname{Lip}_1(X,\mathbb{R})$ is the set of all $1$-Lipschitz functions + $f:X\to\mathbb{R}$, and $f_*\mu_X$ is the pushforward measure on $\mathbb{R}$. + \end{block} + + \vspace{0.4em} + \begin{itemize} + \item Each $1$-Lipschitz function $f$ is viewed as an \textbf{observable} on $X$. + \item The pushforward measure $f_*\mu_X$ is the distribution of the values of that observable. + \item If $\obdiam_{\mathbb{R}}(X;-\kappa)$ is small, then \emph{every} $1$-Lipschitz observable is strongly concentrated. + \end{itemize} + \end{frame} \begin{frame}{A Geometric Consequence} + In this thesis, entropy functions are used as concrete observables to estimate observable diameter, and the Hopf fibration helps transfer information between $S^{2n+1}$ and $\mathbb{C}P^n$. + \vspace{0.4em} \begin{block}{Projective-space estimate} For $0<\kappa<1$, $$