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