Express the epigraph of a parabola as a set of linear inequalities











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Consider the following problem, from the Book Geometry by Prasolov and Tikhomirov:




Specify the following sets by systems of linear inequalities:
$x_2ge x_1^2$ (the "epigraph" of a parabola).




I am puzzled. Since the parabola is a quadratic function, how can I express this set with linear inequalities?



Somehow this should be possible, because there is a subsequent exercise which asks to describe the following set:



${(x_1,x_2) | ; x_1x_2ge 1,; x_1ge0, ; x_2ge 0}$



as a system of linear inequalities.










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  • 1




    Consider it as the intersection of (the set of) epigraphs of all tangents.
    – metamorphy
    Nov 25 at 20:09












  • But wouldn't this require infinitely many (uncountably many) equations? Something like: $x_1ge 2x_2 + q$, $qinmathbb{R}$?
    – J. D.
    Nov 25 at 20:19






  • 1




    Of course it requires an infinite system of inequalities (technically, "countably many" is sufficient here - if we take tangents at points of some dense countable subset of the boundary). It is not asked to find finite systems - and that cannot be done.
    – metamorphy
    Nov 26 at 1:20












  • Ok, I see. I consider it a bit a misleading problem, but I guess that the authors just wanted to underline how linear inequalities can be used to approximate the description of nonlinear convex sets.
    – J. D.
    Nov 26 at 9:12















up vote
0
down vote

favorite












Consider the following problem, from the Book Geometry by Prasolov and Tikhomirov:




Specify the following sets by systems of linear inequalities:
$x_2ge x_1^2$ (the "epigraph" of a parabola).




I am puzzled. Since the parabola is a quadratic function, how can I express this set with linear inequalities?



Somehow this should be possible, because there is a subsequent exercise which asks to describe the following set:



${(x_1,x_2) | ; x_1x_2ge 1,; x_1ge0, ; x_2ge 0}$



as a system of linear inequalities.










share|cite|improve this question


















  • 1




    Consider it as the intersection of (the set of) epigraphs of all tangents.
    – metamorphy
    Nov 25 at 20:09












  • But wouldn't this require infinitely many (uncountably many) equations? Something like: $x_1ge 2x_2 + q$, $qinmathbb{R}$?
    – J. D.
    Nov 25 at 20:19






  • 1




    Of course it requires an infinite system of inequalities (technically, "countably many" is sufficient here - if we take tangents at points of some dense countable subset of the boundary). It is not asked to find finite systems - and that cannot be done.
    – metamorphy
    Nov 26 at 1:20












  • Ok, I see. I consider it a bit a misleading problem, but I guess that the authors just wanted to underline how linear inequalities can be used to approximate the description of nonlinear convex sets.
    – J. D.
    Nov 26 at 9:12













up vote
0
down vote

favorite









up vote
0
down vote

favorite











Consider the following problem, from the Book Geometry by Prasolov and Tikhomirov:




Specify the following sets by systems of linear inequalities:
$x_2ge x_1^2$ (the "epigraph" of a parabola).




I am puzzled. Since the parabola is a quadratic function, how can I express this set with linear inequalities?



Somehow this should be possible, because there is a subsequent exercise which asks to describe the following set:



${(x_1,x_2) | ; x_1x_2ge 1,; x_1ge0, ; x_2ge 0}$



as a system of linear inequalities.










share|cite|improve this question













Consider the following problem, from the Book Geometry by Prasolov and Tikhomirov:




Specify the following sets by systems of linear inequalities:
$x_2ge x_1^2$ (the "epigraph" of a parabola).




I am puzzled. Since the parabola is a quadratic function, how can I express this set with linear inequalities?



Somehow this should be possible, because there is a subsequent exercise which asks to describe the following set:



${(x_1,x_2) | ; x_1x_2ge 1,; x_1ge0, ; x_2ge 0}$



as a system of linear inequalities.







geometry






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share|cite|improve this question











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asked Nov 25 at 19:30









J. D.

1858




1858








  • 1




    Consider it as the intersection of (the set of) epigraphs of all tangents.
    – metamorphy
    Nov 25 at 20:09












  • But wouldn't this require infinitely many (uncountably many) equations? Something like: $x_1ge 2x_2 + q$, $qinmathbb{R}$?
    – J. D.
    Nov 25 at 20:19






  • 1




    Of course it requires an infinite system of inequalities (technically, "countably many" is sufficient here - if we take tangents at points of some dense countable subset of the boundary). It is not asked to find finite systems - and that cannot be done.
    – metamorphy
    Nov 26 at 1:20












  • Ok, I see. I consider it a bit a misleading problem, but I guess that the authors just wanted to underline how linear inequalities can be used to approximate the description of nonlinear convex sets.
    – J. D.
    Nov 26 at 9:12














  • 1




    Consider it as the intersection of (the set of) epigraphs of all tangents.
    – metamorphy
    Nov 25 at 20:09












  • But wouldn't this require infinitely many (uncountably many) equations? Something like: $x_1ge 2x_2 + q$, $qinmathbb{R}$?
    – J. D.
    Nov 25 at 20:19






  • 1




    Of course it requires an infinite system of inequalities (technically, "countably many" is sufficient here - if we take tangents at points of some dense countable subset of the boundary). It is not asked to find finite systems - and that cannot be done.
    – metamorphy
    Nov 26 at 1:20












  • Ok, I see. I consider it a bit a misleading problem, but I guess that the authors just wanted to underline how linear inequalities can be used to approximate the description of nonlinear convex sets.
    – J. D.
    Nov 26 at 9:12








1




1




Consider it as the intersection of (the set of) epigraphs of all tangents.
– metamorphy
Nov 25 at 20:09






Consider it as the intersection of (the set of) epigraphs of all tangents.
– metamorphy
Nov 25 at 20:09














But wouldn't this require infinitely many (uncountably many) equations? Something like: $x_1ge 2x_2 + q$, $qinmathbb{R}$?
– J. D.
Nov 25 at 20:19




But wouldn't this require infinitely many (uncountably many) equations? Something like: $x_1ge 2x_2 + q$, $qinmathbb{R}$?
– J. D.
Nov 25 at 20:19




1




1




Of course it requires an infinite system of inequalities (technically, "countably many" is sufficient here - if we take tangents at points of some dense countable subset of the boundary). It is not asked to find finite systems - and that cannot be done.
– metamorphy
Nov 26 at 1:20






Of course it requires an infinite system of inequalities (technically, "countably many" is sufficient here - if we take tangents at points of some dense countable subset of the boundary). It is not asked to find finite systems - and that cannot be done.
– metamorphy
Nov 26 at 1:20














Ok, I see. I consider it a bit a misleading problem, but I guess that the authors just wanted to underline how linear inequalities can be used to approximate the description of nonlinear convex sets.
– J. D.
Nov 26 at 9:12




Ok, I see. I consider it a bit a misleading problem, but I guess that the authors just wanted to underline how linear inequalities can be used to approximate the description of nonlinear convex sets.
– J. D.
Nov 26 at 9:12















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