# linear algebra – How many inequalities do I necessity to make sure a exclusive resolution? Answer

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## linear algebra – How many inequalities do I necessity to make sure a exclusive resolution?

Suppose there’s a ‘strictly constructive’ vector $$vec x in (0,1) ^ ok$$that lies on the likelihood simplex $$sum_i x_i = 1$$. What is the smallest variety of inequalities of figure $$vec g_i ^ T vec x geq 0$$ required to make positive the answer clique is a singleton vector? Note that there is no such thing as a restriction on $${ vec g_i }$$ so one can select the ‘greatest’ workable vectors (it ought to breathe constant within the sense that there has to breathe an answer on the simplex with constructive elements).

I cerebrate the respond would breathe $$2 (k-1)$$ – I do know that it’s undoubtedly sufficient, however apparently I can’t display that it’s too needful. Construction for sufficiency – take pairs of $$vec g_i, – vec g_i$$ so primarily you’ll have $$k-1$$ Shape equations $$vec x ^ T vec g_i = 0$$ and $$sum_i x_i = 1$$. We have $$ok$$ Equations in order that this has a exclusive resolution and it isn’t troublesome to do that by apt altenative of $$vec g_i$$, we will add constructive elements to the answer.

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