Given a sequence $a_n$, construct the largest minorant $b_n$, i.e. $b_n \leq a_n$, that is convex.
A natural candidate is $$ b_n = \inf \left\{\frac{(r-n)a_l + (n-l)a_r}{r-l} \mid l \leq n < r\right\} $$ assuming that this is well-defined. This is clearly the case when the original sequence is bounded from below.
The values in the infimum are precisely the values of the lines joining $a_l$ and $a_r$ at $n$. Any convex minorant should at least fulfill this condition.
However, why is this regularization convex?
Note that this question solves Logarithmically Convex Minorant Sequence.
