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 $n>3$.






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    $begingroup$


    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







    share|cite|improve this question









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      2








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      $begingroup$


      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







      share|cite|improve this question









      $endgroup$




      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 $n>3$.






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      metric-spaces






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      asked Jan 14 at 20:06









      Rafael DeigaRafael Deiga

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