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Let $n \ge 1$ be an integer, $p \in [1,\infty]$, and $P$ and $Q$ be two (probability) measures on the metric space space $X=(\mathbb R^n,\ell_p)$ which have densities w.r.t the Lebesgue measure on $X$, and let $\varepsilon \ge 0$.

Question

  • (Perhaps with further restrictions on $P$ and $Q$) Are there any interesting estimates (upper bounds are better) for the quantity

$$ \gamma := \inf_{x,x' \in \mathbb B_X}TV(P+\varepsilon x,Q+\varepsilon x'), $$ where $\mathbb B_X$ is the unit ball of $X$, and $P+x$ is the pushforward of $P$ by the translation specified by the vector $x \in X$ ?

  • Same question with $x' = \pm x$ in the constraints.

Of course, the bound $\gamma \le TV(P,Q)$ is trivial and uninteresting (for example, it is independent of the data $\varepsilon$).

N.B.: I'm particularly interested in the cases $p \in \{1,2,\infty\}$.

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  • $\begingroup$ What does it mean to be a measure "with density" on a Banach space? A density with respect to what reference measure? $\endgroup$
    – Nate Eldredge
    Commented Nov 27, 2019 at 12:51
  • $\begingroup$ Oops! OK, I naively assumed there would be an extension of the notion of "Lebesgue measure" to infinite-dimensional Banach spaces. Thus by "densities", I meant densities w.r.t to such a Leb measure. It turns out that no such extension exists en.wikipedia.org/wiki/…. $\endgroup$
    – dohmatob
    Commented Nov 27, 2019 at 13:54
  • $\begingroup$ I've modified the question. Thanks for any constructive feedback. $\endgroup$
    – dohmatob
    Commented Nov 27, 2019 at 14:08

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