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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:

  1. Has work been done on this? If so, what is known?
  2. 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.

  1. Fix a vector field $v(x)$ on $\mathbf{R}^d$
  2. 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 $0 \leqslant t \leqslant T$.

  1. 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.

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    $\begingroup$ For usual Wasserstein distances you need that the measures have equal mass (and yours have different mass, you could normalize by the d-1-dimensional volume, though). Why not use the Hausdorff distance of the level sets? $\endgroup$ – Dirk Feb 19 '19 at 16:42
  • $\begingroup$ @Dirk Thanks for the comment - I did intend for them to be normalised, and hence to be probability measures. I'm interested in measuring the distance between them as probability measures, and the coupling perspective is useful for my application. $\endgroup$ – πr8 Feb 19 '19 at 16:47

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