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
edited Jan 8 at 4:44
Eric Wofsey
183k13209337
asked Jan 8 at 4:36
enthdegreeenthdegree
2,61821335
$endgroup$
add a comment |
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
edited Jan 8 at 4:44
Eric Wofsey
183k13209337
asked Jan 8 at 4:36
enthdegreeenthdegree
2,61821335
$endgroup$
$begingroup$
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
add a comment |
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
edited Jan 8 at 4:44
Eric Wofsey
183k13209337
asked Jan 8 at 4:36
enthdegreeenthdegree
2,61821335
$endgroup$
$A-B:={c:B+csubseteq A}$, 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
edited Jan 8 at 4:44
Eric Wofsey
183k13209337
asked Jan 8 at 4:36
enthdegreeenthdegree
2,61821335
edited Jan 8 at 4:44
Eric Wofsey
183k13209337
asked Jan 8 at 4:36
enthdegreeenthdegree
2,61821335
edited Jan 8 at 4:44
Eric Wofsey
183k13209337
edited Jan 8 at 4:44
Eric Wofsey
183k13209337
edited Jan 8 at 4:44
Eric Wofsey
183k13209337
183k13209337
asked Jan 8 at 4:36
enthdegreeenthdegree
2,61821335
asked Jan 8 at 4:36
enthdegreeenthdegree
2,61821335
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
add a comment |
$begingroup$
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
add a comment |
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$begingroup$
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