update>

latex/chapters/chap2.tex

@@ -9,6 +9,14 @@
 
 \chapter{Levy's family and observable diameters}
 
+\begin{abstract}
+The concentration of measure phenomenon come with important notation called the observable diameter, being the smallest number $D$ such that forall 1-lipschitz function $f$ from $X$ to $\R$, where $X$ is a metric measure space, except on a set with measure $\kappa$, the value of $f$ is concentrated in interval with length $D$.
+
+From Hayden's work, we know that a random pure state bipartite system, the entropy is nearly the maximal and the entropy function has Lipschitz constant upper bounded by $\sqrt{8}\log_2(d_A)$ for $d_A\geq 3$. We can use the entropy function as a proxy for the observable diameter.
+
+Altogether, we experimenting how entropy concentration reflects the geometry of high dimensional spheres, generic bipartite projective state spaces, and symmetric quantum state manifolds.
+\end{abstract}
+
 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.
@@ -113,7 +121,7 @@ Few additional proposition in \cite{shioya2014metricmeasuregeometry} will help u
  
   and
   $$
-  \diam(g_*\mu_X;1-\kappa)\leq \diam((f\circ g)_*\mu_Y;1-\kappa)\leq \obdiam(Y;1-\kappa)
+  \diam(g_*\mu_X;1-\kappa)\leq \diam((f\circ g)_*\mu_Y;1-\kappa)\leq \obdiam(Y;-\kappa)
   $$
 \end{proof}