N - 1 Equal Value Principle
The n - 1 Equal Value Principle, or n - 1 EV for short, is a useful fact regarding the optimization of sums of values of twice continuously differentiable functions with only one inflection point when the sum of the input values is constant.
Statement
Say we have where
is twice continuously differentiable and only has one inflection point. When
are real numbers such that
for some constant
,
is maximized or minimized only if
of the
-values are equal.
Proof
The following proof is adapted from Mildorf's Olympiad Inqeualities.
Give me any sequence of real numbers
and I show that my sequence, with
equal terms, gives a greater or equal result. (Obviously, similar reasoning applies when minimizing the sum.)
WLOG assume that for some value
,
's inflection point,
is convex for values below and at
and concave otherwise and at
. Also assume that
are non-increasing. Now, define
so that
.
Since
and
is convex below and at
, we can apply Karamata's Inequality to obtain:
Since
and f is concave (as opposed to convex) above and at
, we can apply Karamata's Inequality once more to obtain:
Finally, by the transitive property,
as desired.
The following video by Evan Chen may be a helpful aid in understanding this technique: https://www.youtube.com/watch?v=Kkhuk8GiuOU