# Bounds on $\inf_{x,x' \in \mathbb B_X}TV(P+x,Q+x')$, where $P$ and $Q$ are distributions with density on the space $X=(\mathbb R^n,\ell_p)$

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 $$p \in \{1,2,\infty\}$$.

• What does it mean to be a measure "with density" on a Banach space? A density with respect to what reference measure? – Nate Eldredge Nov 27 at 12:51
• 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/…. – dohmatob Nov 27 at 13:54
• I've modified the question. Thanks for any constructive feedback. – dohmatob Nov 27 at 14:08