Convex function's monotonicity?
Let $F:[0, infty)^nrightarrow [0, infty)$ be a convex function and $F(x_1,x_2,...,x_n)=0$ if and only if $x_i=0$ for $i=1,...,n$.
Is $F$ a nondecreasing function, i.e if $x_ileq y_i$ for $i=1,...,n$, then $F(x_1,x_2,...,x_n)leq F(y_1,y_2,...,y_n)$?
If not do you have any counterexample in mind?
Thank you in advance
real-analysis functional-analysis functions convex-analysis
edited Nov 27 at 15:02
gerw
19k11133
asked Nov 27 at 12:32
Motaka
229111
add a comment |
Let $F:[0, infty)^nrightarrow [0, infty)$ be a convex function and $F(x_1,x_2,...,x_n)=0$ if and only if $x_i=0$ for $i=1,...,n$.
Is $F$ a nondecreasing function, i.e if $x_ileq y_i$ for $i=1,...,n$, then $F(x_1,x_2,...,x_n)leq F(y_1,y_2,...,y_n)$?
If not do you have any counterexample in mind?
Thank you in advance
real-analysis functional-analysis functions convex-analysis
edited Nov 27 at 15:02
gerw
19k11133
asked Nov 27 at 12:32
Motaka
229111
add a comment |
Let $F:[0, infty)^nrightarrow [0, infty)$ be a convex function and $F(x_1,x_2,...,x_n)=0$ if and only if $x_i=0$ for $i=1,...,n$.
Is $F$ a nondecreasing function, i.e if $x_ileq y_i$ for $i=1,...,n$, then $F(x_1,x_2,...,x_n)leq F(y_1,y_2,...,y_n)$?
If not do you have any counterexample in mind?
Thank you in advance
real-analysis functional-analysis functions convex-analysis
edited Nov 27 at 15:02
gerw
19k11133
asked Nov 27 at 12:32
Motaka
229111
Let $F:[0, infty)^nrightarrow [0, infty)$ be a convex function and $F(x_1,x_2,...,x_n)=0$ if and only if $x_i=0$ for $i=1,...,n$.
Is $F$ a nondecreasing function, i.e if $x_ileq y_i$ for $i=1,...,n$, then $F(x_1,x_2,...,x_n)leq F(y_1,y_2,...,y_n)$?
If not do you have any counterexample in mind?
Thank you in advance
real-analysis functional-analysis functions convex-analysis
real-analysis functional-analysis functions convex-analysis
edited Nov 27 at 15:02
gerw
19k11133
asked Nov 27 at 12:32
Motaka
229111
edited Nov 27 at 15:02
gerw
19k11133
asked Nov 27 at 12:32
Motaka
229111
edited Nov 27 at 15:02
gerw
19k11133
edited Nov 27 at 15:02
gerw
19k11133
edited Nov 27 at 15:02
gerw
19k11133
19k11133
asked Nov 27 at 12:32
Motaka
229111
asked Nov 27 at 12:32
Motaka
229111
asked Nov 27 at 12:32
Motaka
229111
229111
add a comment |
add a comment |
1 Answer
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Not necessarily, for example take $F(x) = (x_1-x_2)^2+0.1x_2$, $x=(1,0)$, $y=(2,2)$.
answered Nov 27 at 13:54
LinAlg
8,2861521
1
@Motaka thank you, I have updated the example.
– LinAlg
Nov 27 at 16:31
add a comment |
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1 Answer
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1 Answer
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Not necessarily, for example take $F(x) = (x_1-x_2)^2+0.1x_2$, $x=(1,0)$, $y=(2,2)$.
answered Nov 27 at 13:54
LinAlg
8,2861521
1
@Motaka thank you, I have updated the example.
– LinAlg
Nov 27 at 16:31
add a comment |
Not necessarily, for example take $F(x) = (x_1-x_2)^2+0.1x_2$, $x=(1,0)$, $y=(2,2)$.
answered Nov 27 at 13:54
LinAlg
8,2861521
1
@Motaka thank you, I have updated the example.
– LinAlg
Nov 27 at 16:31
add a comment |
Not necessarily, for example take $F(x) = (x_1-x_2)^2+0.1x_2$, $x=(1,0)$, $y=(2,2)$.
answered Nov 27 at 13:54
LinAlg
8,2861521
Not necessarily, for example take $F(x) = (x_1-x_2)^2+0.1x_2$, $x=(1,0)$, $y=(2,2)$.
answered Nov 27 at 13:54
LinAlg
8,2861521
edited Nov 27 at 16:31
answered Nov 27 at 13:54
LinAlg
8,2861521
answered Nov 27 at 13:54
LinAlg
8,2861521
answered Nov 27 at 13:54
LinAlg
8,2861521
8,2861521
1
@Motaka thank you, I have updated the example.
– LinAlg
Nov 27 at 16:31
add a comment |
1
@Motaka thank you, I have updated the example.
– LinAlg
Nov 27 at 16:31
1
1
@Motaka thank you, I have updated the example.
– LinAlg
Nov 27 at 16:31
@Motaka thank you, I have updated the example.
– LinAlg
Nov 27 at 16:31
add a comment |
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