IGW curves

By Lemma $3.2$, the $W_2$ geodesic is a valid $\texttt{IGW}$-Lipschitz curve connecting $\mu_0,\mu_1$, hence $\texttt{Lip}{\texttt{IGW}}([0,1],\mathcal{P}_2(\mathbb{R}^d)) \neq \phi$ and $d{\texttt{IGW}} < \infty$ always. Consider now a sequence of curves ${\rho_n}{n\in N}$, reparametrized to be uniformly Lipschitz with constant $\text{Lip}(\rho_n) = \mathcal{l}{\text{IGW}}(\rho_n) < 2d_{\text{IGW}}(\mu_0,\mu_1)$, such that $l$

Lower semicontinuity:

A function $f: X \rightarrow \mathbb{R}$ is called lower semi-continous at a point $x_0 \in X$ if for every real $y < f(x_0)$ there exists a neighborhood $U$ of $x_0$ such that $f(x) > y$ for all $x \in U$. Equivalently, $f$ if lower semi-continuous at $x_0$ if and only if, \(\liminf_{x \rightarrow x_0} f(x)\ge f(x_0)\) where $\liminf$ is the limit inferior of the function $f$ at point $x_0$.

Lemma 4.1: For any $\mu \in \mathcal{P}_2(\mathbb{R}^d)$, we have:

(i): $\forall r > 0$, $B_{IGW}(\mu,r)$ is weakly compact, i.e. any sequence has a weakly convergent subsequence.

(ii): $IGW(\mu, \nu)$ is weakly l.s.c. in $\nu$.