Bound on volume of $A-B$ according to Minkowski












3












$begingroup$


$A-B:={c:B+csubseteq A}$, in, say, Euclidean space.



I think that if $A-B$ is not the universe, then $vol(A-B)leq vol(A)$ (If $B$ has one point then the inequality is immediate, adding more points to $B$ can only reduce $vol(A-B)$ by set theoretic consideration.) Is there anything tighter?






real-analysis geometry volume







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$endgroup$












  • $begingroup$
    Your argument is correct. Also, equality holds when $B$ is a singleton. What do you mean by 'tighter'?
    $endgroup$
    – Kavi Rama Murthy
    Jan 8 at 5:38






  • 1




    $begingroup$
    Beware that you use a standard notation in a non-standard way. The conventional definition (which, I believe, originates from Minkowski's work) is $A-B={a-bcolon ain A, bin B}$.
    $endgroup$
    – W-t-P
    Jan 8 at 10:11










  • $begingroup$
    Alright thanks for the information. I am taking mine from the wiki definition.
    $endgroup$
    – enthdegree
    Jan 8 at 17:14
















3












$begingroup$


$A-B:={c:B+csubseteq A}$, in, say, Euclidean space.



I think that if $A-B$ is not the universe, then $vol(A-B)leq vol(A)$ (If $B$ has one point then the inequality is immediate, adding more points to $B$ can only reduce $vol(A-B)$ by set theoretic consideration.) Is there anything tighter?






real-analysis geometry volume







share|cite|improve this question











$endgroup$












  • $begingroup$
    Your argument is correct. Also, equality holds when $B$ is a singleton. What do you mean by 'tighter'?
    $endgroup$
    – Kavi Rama Murthy
    Jan 8 at 5:38






  • 1




    $begingroup$
    Beware that you use a standard notation in a non-standard way. The conventional definition (which, I believe, originates from Minkowski's work) is $A-B={a-bcolon ain A, bin B}$.
    $endgroup$
    – W-t-P
    Jan 8 at 10:11










  • $begingroup$
    Alright thanks for the information. I am taking mine from the wiki definition.
    $endgroup$
    – enthdegree
    Jan 8 at 17:14














3












3








3


1



$begingroup$


$A-B:={c:B+csubseteq A}$, in, say, Euclidean space.



I think that if $A-B$ is not the universe, then $vol(A-B)leq vol(A)$ (If $B$ has one point then the inequality is immediate, adding more points to $B$ can only reduce $vol(A-B)$ by set theoretic consideration.) Is there anything tighter?






real-analysis geometry volume







share|cite|improve this question











$endgroup$




$A-B:={c:B+csubseteq A}$, in, say, Euclidean space.



I think that if $A-B$ is not the universe, then $vol(A-B)leq vol(A)$ (If $B$ has one point then the inequality is immediate, adding more points to $B$ can only reduce $vol(A-B)$ by set theoretic consideration.) Is there anything tighter?






real-analysis geometry volume




real-analysis geometry volume






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Jan 8 at 4:44









Eric Wofsey

183k13209337




183k13209337










asked Jan 8 at 4:36









enthdegreeenthdegree

2,61821335




2,61821335












  • $begingroup$
    Your argument is correct. Also, equality holds when $B$ is a singleton. What do you mean by 'tighter'?
    $endgroup$
    – Kavi Rama Murthy
    Jan 8 at 5:38






  • 1




    $begingroup$
    Beware that you use a standard notation in a non-standard way. The conventional definition (which, I believe, originates from Minkowski's work) is $A-B={a-bcolon ain A, bin B}$.
    $endgroup$
    – W-t-P
    Jan 8 at 10:11










  • $begingroup$
    Alright thanks for the information. I am taking mine from the wiki definition.
    $endgroup$
    – enthdegree
    Jan 8 at 17:14


















  • $begingroup$
    Your argument is correct. Also, equality holds when $B$ is a singleton. What do you mean by 'tighter'?
    $endgroup$
    – Kavi Rama Murthy
    Jan 8 at 5:38






  • 1




    $begingroup$
    Beware that you use a standard notation in a non-standard way. The conventional definition (which, I believe, originates from Minkowski's work) is $A-B={a-bcolon ain A, bin B}$.
    $endgroup$
    – W-t-P
    Jan 8 at 10:11










  • $begingroup$
    Alright thanks for the information. I am taking mine from the wiki definition.
    $endgroup$
    – enthdegree
    Jan 8 at 17:14
















$begingroup$
Your argument is correct. Also, equality holds when $B$ is a singleton. What do you mean by 'tighter'?
$endgroup$
– Kavi Rama Murthy
Jan 8 at 5:38




$begingroup$
Your argument is correct. Also, equality holds when $B$ is a singleton. What do you mean by 'tighter'?
$endgroup$
– Kavi Rama Murthy
Jan 8 at 5:38




1




1




$begingroup$
Beware that you use a standard notation in a non-standard way. The conventional definition (which, I believe, originates from Minkowski's work) is $A-B={a-bcolon ain A, bin B}$.
$endgroup$
– W-t-P
Jan 8 at 10:11




$begingroup$
Beware that you use a standard notation in a non-standard way. The conventional definition (which, I believe, originates from Minkowski's work) is $A-B={a-bcolon ain A, bin B}$.
$endgroup$
– W-t-P
Jan 8 at 10:11












$begingroup$
Alright thanks for the information. I am taking mine from the wiki definition.
$endgroup$
– enthdegree
Jan 8 at 17:14




$begingroup$
Alright thanks for the information. I am taking mine from the wiki definition.
$endgroup$
– enthdegree
Jan 8 at 17:14










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