What does a segment in the plane as a metric space defined by a $p$-norm look like?

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In the metric space $mathbb R^2$ with the metric $d$ defined by $d(x,y)= (|x_1-y_1|^p+|x_2-y_2|^p)^1/p$, where $pgt1$ is a real number, like what does the set of all $min mathbb R^2$ with $d(a,m)+d(m,b)=d(a,b)$ look, where $a$ and $b$ are two arbitrary points of $mathbb R^2$?
I think for all $p$’s it is a straight line segment as in the obvious case $p=2$ but I do not know how to deal with the other values of $p$.






real-analysis metric-spaces







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  • 3




    Check when equality holds in Minkowski's inequality
    – Michal Adamaszek
    Sep 4 at 10:07










  • Yes Minkowski’s inequality shows it’s a line segment, thanks @MichalAdamaszek
    – Selflearner
    Sep 4 at 10:20














up vote
5
down vote

favorite












In the metric space $mathbb R^2$ with the metric $d$ defined by $d(x,y)= (|x_1-y_1|^p+|x_2-y_2|^p)^1/p$, where $pgt1$ is a real number, like what does the set of all $min mathbb R^2$ with $d(a,m)+d(m,b)=d(a,b)$ look, where $a$ and $b$ are two arbitrary points of $mathbb R^2$?
I think for all $p$’s it is a straight line segment as in the obvious case $p=2$ but I do not know how to deal with the other values of $p$.






real-analysis metric-spaces







share|cite|improve this question

















  • 3




    Check when equality holds in Minkowski's inequality
    – Michal Adamaszek
    Sep 4 at 10:07










  • Yes Minkowski’s inequality shows it’s a line segment, thanks @MichalAdamaszek
    – Selflearner
    Sep 4 at 10:20












up vote
5
down vote

favorite









up vote
5
down vote

favorite











In the metric space $mathbb R^2$ with the metric $d$ defined by $d(x,y)= (|x_1-y_1|^p+|x_2-y_2|^p)^1/p$, where $pgt1$ is a real number, like what does the set of all $min mathbb R^2$ with $d(a,m)+d(m,b)=d(a,b)$ look, where $a$ and $b$ are two arbitrary points of $mathbb R^2$?
I think for all $p$’s it is a straight line segment as in the obvious case $p=2$ but I do not know how to deal with the other values of $p$.






real-analysis metric-spaces







share|cite|improve this question













In the metric space $mathbb R^2$ with the metric $d$ defined by $d(x,y)= (|x_1-y_1|^p+|x_2-y_2|^p)^1/p$, where $pgt1$ is a real number, like what does the set of all $min mathbb R^2$ with $d(a,m)+d(m,b)=d(a,b)$ look, where $a$ and $b$ are two arbitrary points of $mathbb R^2$?
I think for all $p$’s it is a straight line segment as in the obvious case $p=2$ but I do not know how to deal with the other values of $p$.






real-analysis metric-spaces




real-analysis metric-spaces






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




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asked Sep 4 at 9:51









Selflearner

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  • 3




    Check when equality holds in Minkowski's inequality
    – Michal Adamaszek
    Sep 4 at 10:07










  • Yes Minkowski’s inequality shows it’s a line segment, thanks @MichalAdamaszek
    – Selflearner
    Sep 4 at 10:20












  • 3




    Check when equality holds in Minkowski's inequality
    – Michal Adamaszek
    Sep 4 at 10:07










  • Yes Minkowski’s inequality shows it’s a line segment, thanks @MichalAdamaszek
    – Selflearner
    Sep 4 at 10:20







3




3




Check when equality holds in Minkowski's inequality
– Michal Adamaszek
Sep 4 at 10:07




Check when equality holds in Minkowski's inequality
– Michal Adamaszek
Sep 4 at 10:07












Yes Minkowski’s inequality shows it’s a line segment, thanks @MichalAdamaszek
– Selflearner
Sep 4 at 10:20




Yes Minkowski’s inequality shows it’s a line segment, thanks @MichalAdamaszek
– Selflearner
Sep 4 at 10:20















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