Express the epigraph of a parabola as a set of linear inequalities
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
asked Nov 25 at 19:30
J. D.
1858
add a comment |
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
asked Nov 25 at 19:30
J. D.
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
add a comment |
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
asked Nov 25 at 19:30
J. D.
1858
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
geometry
asked Nov 25 at 19:30
J. D.
1858
asked Nov 25 at 19:30
J. D.
1858
asked Nov 25 at 19:30
J. D.
1858
asked Nov 25 at 19:30
J. D.
1858
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
add a comment |
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
add a comment |
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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