Suppose I have a well-behaved, strictly convex function $f : \mathbf{R}^d \to [0, \infty)$, and assume that $f$ has its unique minimiser at $x = 0$, with $f(0) = 0$.
For $y > 0$, I define the level of set of $f$ at $y$ by
$$L_y = \{x \in \mathbb{R}^d: f(x) = y \}.$$
Endow $L_y$ with (normalised) $(d-1)$-dimensional Hausdorff measure, and call the resulting probability measure $\mu_y$.
For $y_1, y_2 > 0$, I would like to be able to bound the distance between $\mu_{y_1}$ and $\mu_{y_2}$. Since these measures all have disjoint support, a natural choice of metric would be a transport distance, e.g. a Wasserstein distance $\mathcal{W}_p$. I would like to have a bound of the form, e.g.
$$\mathcal{W}_p (\mu_{y_1}, \mu_{y_2}) \leqslant \omega(y_1, y_2)$$
where $\omega$ is some explicit modulus of continuity, e.g. $\omega(y_1, y_2) = C \|y_1 - y_2\|^\alpha$, for some $C, \alpha > 0$.
My questions are effectively:
- Has work been done on this? If so, what is known?
- What sort of additional conditions on $f$ would be useful for establishing bounds of this form?
A related problem which I am also interested involves the same question, but replacing 'level sets' with 'flows of an ODE', e.g.
- Fix a vector field $v(x)$ on $\mathbf{R}^d$
- For $x_0 \in \mathbf{R}^d$, $T > 0$, let $F_{x_0,T}$ be the path traced out in $\mathbf{R}^d$ by the ODE
\begin{align} \dot{x} &= v(x) \\ x(0) &= x_0 \quad \end{align}
for $0 \leqslant t \leqslant T$.
- Endow $F_{x_0,T}$ with (normalised) $(d-1)$-dimensional Hausdorff measure, and call the resulting probability measure $\pi_{x_0, T}$.
and then bounding transport distances between $\pi_{x_1, T}$, $\pi_{x_2, T}$, for different initial values $x_1, x_2 \in \mathbf{R}^d$.
I don't have a strong intuition for whether this is an easier or harder problem than the level set problem, but thought it would be worth including in any case.
