/sci/ - Science & Math

Fix some real numbers [math]a_1, a_2, \ldots, a_N \in \mathbb{R}[/math]. Consider also some weights [math]c_i \geq 0, i = 1, \ldots, N[/math], that sum to 1, [math]\sum_{i=1}^N c_i = 1[/math]. Define [math]f(c) = \left( \sum_{i=1}^N c_i a_i \right)^2[/math] as (the square of) the weighted average of [math]a[/math].

Now suppose I want to generalize to "weights" that can be negative. Call them [math]c_i' \in \mathbb{R}[/math]. We normalize them as follows: [math]\sum_{i=1}^N |c_i'| \leq 1[/math]. The set of all [math]c'[/math] is clearly a superset of all [math]c[/math]. My question is, for any [math]c'[/math], does there exist a [math]c[/math] such that [math]f(c') \leq f(c)[/math] (for any fixed [math]a[/math])?