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Maximum number of elements of $X subset mathbb{R}^n$ such that the discrete metric in $X$ is induced by the...
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What is the maximum number of elements of $X subset mathbb{R}^n$ such that the discrete metric in $X$ is induced by the eucliean one in $mathbb{R}^n$?
It's easy to verify that if $n=1$, then the answer is $2$. The case $n = 2$ has already been solved (What is the maximum number of points that a subspace $X subset mathbb{R}²$ can have so that $mathbb{R}²$ induces the discrete metric in $X$?), whose answer is 3. In similar way with solid geomtry, one can prove that if $n=3$, then the answer is 4. So, a reasonable conjecture is $n+1$. However, I couldn't find a solution for $n>3$.
metric-spaces
asked Jan 14 at 20:06
Rafael DeigaRafael Deiga
672311
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add a comment |
$begingroup$
What is the maximum number of elements of $X subset mathbb{R}^n$ such that the discrete metric in $X$ is induced by the eucliean one in $mathbb{R}^n$?
It's easy to verify that if $n=1$, then the answer is $2$. The case $n = 2$ has already been solved (What is the maximum number of points that a subspace $X subset mathbb{R}²$ can have so that $mathbb{R}²$ induces the discrete metric in $X$?), whose answer is 3. In similar way with solid geomtry, one can prove that if $n=3$, then the answer is 4. So, a reasonable conjecture is $n+1$. However, I couldn't find a solution for $n>3$.
metric-spaces
asked Jan 14 at 20:06
Rafael DeigaRafael Deiga
672311
$endgroup$
add a comment |
$begingroup$
What is the maximum number of elements of $X subset mathbb{R}^n$ such that the discrete metric in $X$ is induced by the eucliean one in $mathbb{R}^n$?
It's easy to verify that if $n=1$, then the answer is $2$. The case $n = 2$ has already been solved (What is the maximum number of points that a subspace $X subset mathbb{R}²$ can have so that $mathbb{R}²$ induces the discrete metric in $X$?), whose answer is 3. In similar way with solid geomtry, one can prove that if $n=3$, then the answer is 4. So, a reasonable conjecture is $n+1$. However, I couldn't find a solution for $n>3$.
metric-spaces
asked Jan 14 at 20:06
Rafael DeigaRafael Deiga
672311
$endgroup$
What is the maximum number of elements of $X subset mathbb{R}^n$ such that the discrete metric in $X$ is induced by the eucliean one in $mathbb{R}^n$?
It's easy to verify that if $n=1$, then the answer is $2$. The case $n = 2$ has already been solved (What is the maximum number of points that a subspace $X subset mathbb{R}²$ can have so that $mathbb{R}²$ induces the discrete metric in $X$?), whose answer is 3. In similar way with solid geomtry, one can prove that if $n=3$, then the answer is 4. So, a reasonable conjecture is $n+1$. However, I couldn't find a solution for $n>3$.
metric-spaces
metric-spaces
asked Jan 14 at 20:06
Rafael DeigaRafael Deiga
672311
asked Jan 14 at 20:06
Rafael DeigaRafael Deiga
672311
asked Jan 14 at 20:06
Rafael DeigaRafael Deiga
672311
asked Jan 14 at 20:06
Rafael DeigaRafael Deiga
672311
asked Jan 14 at 20:06
Rafael DeigaRafael Deiga
672311
672311
add a comment |
add a comment |
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