With user10354138' comment, let me explain the setting (in the paper) :
Consider a Busemann function in Riemannian manifold $( \mathbb{R}^n,d)$ with $\mathbb{Z}^n$-isometric action $\ast$ where $d$ is a Riemannian distance.
Further, $|\| x-y\| -d(x,y)|<c$ for all $x,\ y$ and some norm $\|\ \|$. Assume that a linear map $L$ is a support function of $\|\ \|$-unit ball. We define $$ B\colon\mathbb{R}^n\rightarrow \mathbb{R},\ B(x) =\limsup_{d(y_0,y) \rightarrow \infty}\ (L(y)- d(x,y)) $$
where $L\colon \mathbb{R}^n\rightarrow \mathbb{R}$ is a linear map and $y_0$ is any reference point.
Prove that $B$ is $1$-Lipschitz map
Proof : I have a proof about special case :
Here we have a level set $A = \{z \in \mathbb{R}^n | L(z)=n \}$ where $n\rightarrow \infty$. We assume that for $\varepsilon$, there is $n$ s.t. $ y_n,\ Y_n \in A$ and $$ | B(x) - L(y_n) + d(x,y_n) |< \varepsilon $$
$$ |B(X) - L(Y_n) + d(X,Y_n) |< \varepsilon $$
\begin{align*} 0 &<_{{\rm assumption}} B(x)-B(X) \\&= d(X,Y_n) - d(x,y_n) +O(\varepsilon ) \\&= d(X,A)-d(x,A) \\&= d(X,A)- d(x,a),\ {\rm some}\ a\in A \\&\leq d(X,a)-d(x,a) \\ &\leq d(x,X)\end{align*}
Can we prove this in general ?