Characterization of the convex hull in terms of dot product

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I am doing some work with Newton Polytopes and I need something of this style:




Given $v_1,dots,v_nin mathbbR^n$ we have



$$textconv(v_1,dots v_n)=vin mathbbR^nmid min_i=1,dots nxcdot v_ileq xcdot v ;;forall , xin mathbbR^n$$




As the condition defining the set in the left is true for the $v_i$ and is closed under convex combinations then $subseteq$ is easy. My problem is to show that




$$min_i=1,dots nxcdot v_ileq xcdot v ;;forall , xin mathbbR^nimplies vin textconv(v_1,dots v_n)$$




This is trivial for $n=1$ but I can't see it for all $n$.



Any help is appreciated.




linear-algebra convex-analysis convex-optimization convex-hulls




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  • 1




    If v wasn't in the convex hull, you could construct an affine function f so that f(v) < 0, while $f(v_i) geq 0$. (Hyperplane separation theorem.)
    – Lorenzo
    Aug 7 at 18:05










  • It was really easy thinking in that way. Thanks! You can put the hint as an answer if you want.
    – Walter Simon
    Aug 7 at 18:12














up vote
1
down vote

favorite
1












I am doing some work with Newton Polytopes and I need something of this style:




Given $v_1,dots,v_nin mathbbR^n$ we have



$$textconv(v_1,dots v_n)=vin mathbbR^nmid min_i=1,dots nxcdot v_ileq xcdot v ;;forall , xin mathbbR^n$$




As the condition defining the set in the left is true for the $v_i$ and is closed under convex combinations then $subseteq$ is easy. My problem is to show that




$$min_i=1,dots nxcdot v_ileq xcdot v ;;forall , xin mathbbR^nimplies vin textconv(v_1,dots v_n)$$




This is trivial for $n=1$ but I can't see it for all $n$.



Any help is appreciated.




linear-algebra convex-analysis convex-optimization convex-hulls




share|cite|improve this question

















  • 1




    If v wasn't in the convex hull, you could construct an affine function f so that f(v) < 0, while $f(v_i) geq 0$. (Hyperplane separation theorem.)
    – Lorenzo
    Aug 7 at 18:05










  • It was really easy thinking in that way. Thanks! You can put the hint as an answer if you want.
    – Walter Simon
    Aug 7 at 18:12












up vote
1
down vote

favorite
1









up vote
1
down vote

favorite
1






1





I am doing some work with Newton Polytopes and I need something of this style:




Given $v_1,dots,v_nin mathbbR^n$ we have



$$textconv(v_1,dots v_n)=vin mathbbR^nmid min_i=1,dots nxcdot v_ileq xcdot v ;;forall , xin mathbbR^n$$




As the condition defining the set in the left is true for the $v_i$ and is closed under convex combinations then $subseteq$ is easy. My problem is to show that




$$min_i=1,dots nxcdot v_ileq xcdot v ;;forall , xin mathbbR^nimplies vin textconv(v_1,dots v_n)$$




This is trivial for $n=1$ but I can't see it for all $n$.



Any help is appreciated.




linear-algebra convex-analysis convex-optimization convex-hulls




share|cite|improve this question













I am doing some work with Newton Polytopes and I need something of this style:




Given $v_1,dots,v_nin mathbbR^n$ we have



$$textconv(v_1,dots v_n)=vin mathbbR^nmid min_i=1,dots nxcdot v_ileq xcdot v ;;forall , xin mathbbR^n$$




As the condition defining the set in the left is true for the $v_i$ and is closed under convex combinations then $subseteq$ is easy. My problem is to show that




$$min_i=1,dots nxcdot v_ileq xcdot v ;;forall , xin mathbbR^nimplies vin textconv(v_1,dots v_n)$$




This is trivial for $n=1$ but I can't see it for all $n$.



Any help is appreciated.





linear-algebra convex-analysis convex-optimization convex-hulls





share|cite|improve this question












share|cite|improve this question




share|cite|improve this question








edited Aug 7 at 18:15









Rodrigo de Azevedo

12.6k41751




12.6k41751









asked Aug 7 at 17:59









Walter Simon

749




749







  • 1




    If v wasn't in the convex hull, you could construct an affine function f so that f(v) < 0, while $f(v_i) geq 0$. (Hyperplane separation theorem.)
    – Lorenzo
    Aug 7 at 18:05










  • It was really easy thinking in that way. Thanks! You can put the hint as an answer if you want.
    – Walter Simon
    Aug 7 at 18:12












  • 1




    If v wasn't in the convex hull, you could construct an affine function f so that f(v) < 0, while $f(v_i) geq 0$. (Hyperplane separation theorem.)
    – Lorenzo
    Aug 7 at 18:05










  • It was really easy thinking in that way. Thanks! You can put the hint as an answer if you want.
    – Walter Simon
    Aug 7 at 18:12







1




1




If v wasn't in the convex hull, you could construct an affine function f so that f(v) < 0, while $f(v_i) geq 0$. (Hyperplane separation theorem.)
– Lorenzo
Aug 7 at 18:05




If v wasn't in the convex hull, you could construct an affine function f so that f(v) < 0, while $f(v_i) geq 0$. (Hyperplane separation theorem.)
– Lorenzo
Aug 7 at 18:05












It was really easy thinking in that way. Thanks! You can put the hint as an answer if you want.
– Walter Simon
Aug 7 at 18:12




It was really easy thinking in that way. Thanks! You can put the hint as an answer if you want.
– Walter Simon
Aug 7 at 18:12















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