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I would like to know what is the locus of $x \in \Bbb R_+^n$ ($n=2$ would already be fine) defined by

$\sum a_i \cdot x_i$ s.t. $a_i+\epsilon \geq 0$, $\epsilon \in \Bbb R$.

I know that if $\epsilon =0$, it's the conical hull (see here for details) of the data points, but I'd like to generalize to $\epsilon >0$ and $\epsilon <0$

I would also like to get the same in the "convex" case (i.e. when adding the constraint $\sum a_i = 1$). It would result in a kind of "nearly convex"combination. As you can notice, assume that the data $x$ are positive.

Thank you

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  • $\begingroup$ Do you mean the locus of $x \in \mathbb{R}^n_+$ satisfying the equation $\sum a_i \cdot x_i = 0$? I don't know how to define a locus by an expression; only an equation or inequality. I also don't understand the constraint on $a_i+\varepsilon$. Do you mean the locus of $x \in \mathbb{R}^n_+$ satisfying the equation $\sum a_i \cdot x_i = 0$ for every choice of numbers $a_1,\dots,a_n$ so that $a_i+\varepsilon\ge 0$ for any $\varepsilon \in \mathbb{R}$ and for every $i=1,2,\dots,n$? I don't understand. $\endgroup$
    – Ben McKay
    Commented Feb 13, 2019 at 21:39
  • $\begingroup$ What are the "data points"? $\endgroup$
    – Ben McKay
    Commented Feb 13, 2019 at 21:40
  • $\begingroup$ @BenMcKay For example, if $\epsilon =0$, it's obviously a cone ($ \sum a_i x_i | a_i \in R_+, x_i \in R_+^n $). I would like to extend this to the case where rather to have $a_i \geq 0$, one haves $a_i \geq -\epsilon$, $\epsilon >0$ Hope it's clearer ? $\endgroup$
    – MysteryGuy
    Commented Feb 14, 2019 at 6:44

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