Show that for some $\rho>0$
$$ \inf_{|z| \leq \rho} \int_{-\infty}^{\infty} (r \delta(x-k) \land r \delta(x+z-k)) \ \textrm{d}x >0$$
where $r,k,z>0$?
I am not sure how to deal with the ∧ operator and the delta functions. Thanks for your help!
Edit: I solved my own question (see below). The inequality is never satisfied.
I figured out that the inequality is never satisfied.
$$ \inf_{|z| \leq \rho} \int_{-\infty}^{\infty} (r \delta(x-k) \land r \delta(x+z-k)) \ \textrm{d}x >0$$
$$ = \inf_{|z| \leq \rho} \int_{-\infty}^{\infty} \min{(r \delta(x-k), r \delta(x+z-k))} \ \textrm{d}x >0$$
The minimum of two delta functions with always be zero unless they are the centered at the same point, i.e. $z=0$. Then $$\inf_{|z| \leq \rho} \int_{-\infty}^{\infty} (r \delta(x-k) \land r \delta(x+z-k)) \ \textrm{d}x =\inf_{|z| \leq \rho} r\mathbb{1}_z$$ where $$ \mathbb{1}_z = \begin{cases} 1 & z=0 \\ 0 & \text{otherwise}\end{cases}$$
which is always zero.