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7

1

$begingroup$

Let $U$ be the set of all nonempty subsets of $[0,1]$ that are a union of finitely many closed intervals (where an "interval" that is a single point does not count as an interval). Does there exist a function $f:Utimes Urightarrow U$ such that for any $A,Bin U$:

(a) $f(A,B)neq [0,1]$.

(b) $f(A,B)cap Aneqemptyset$ and $f(A,B)cap Bneqemptyset$.

(c) The length (i.e. Lebesgue measure) of $f(X,B)cap A$ is maximized at $X=A$, and the length of $f(A,X)cap B$ is maximized at $X=B$.

Any two of the three conditions can be satisfied:

• $f(A,B)equiv [0,1]$ satisfy (b) and (c).




• $f(A,B)equiv Y$ for any fixed $Yneq [0,1]$ satisfy (a) and (c).




• $f(A,B)$ being any $Yneq [0,1]$ that intersects both $A$ and $B$ satisfy (a) and (b).

Satisfying all three seems to be impossible though.

(Question posted on MO five months ago but still unsolved.)

real-analysis lebesgue-measure

share|cite|improve this question

edited Mar 21 at 5:30

Eric Wofsey

190k14216348

asked Mar 16 at 12:10

pi66pi66

3,9871239

$endgroup$

This question has an open bounty worth +50
reputation from pi66 ending ending at 2019-03-25 12:10:45Z">in 5 hours.

This question has not received enough attention.

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Is $[0,1] in U$?
  $endgroup$
  – Berci
  Mar 16 at 12:30
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @Berci Yes, it is.
  $endgroup$
  – pi66
  Mar 16 at 12:53
  

  
  
  
  

add a comment | 

7

1

$begingroup$

Let $U$ be the set of all nonempty subsets of $[0,1]$ that are a union of finitely many closed intervals (where an "interval" that is a single point does not count as an interval). Does there exist a function $f:Utimes Urightarrow U$ such that for any $A,Bin U$:

(a) $f(A,B)neq [0,1]$.

(b) $f(A,B)cap Aneqemptyset$ and $f(A,B)cap Bneqemptyset$.

(c) The length (i.e. Lebesgue measure) of $f(X,B)cap A$ is maximized at $X=A$, and the length of $f(A,X)cap B$ is maximized at $X=B$.

Any two of the three conditions can be satisfied:

• $f(A,B)equiv [0,1]$ satisfy (b) and (c).




• $f(A,B)equiv Y$ for any fixed $Yneq [0,1]$ satisfy (a) and (c).




• $f(A,B)$ being any $Yneq [0,1]$ that intersects both $A$ and $B$ satisfy (a) and (b).

Satisfying all three seems to be impossible though.

(Question posted on MO five months ago but still unsolved.)

real-analysis lebesgue-measure

share|cite|improve this question

edited Mar 21 at 5:30

Eric Wofsey

190k14216348

asked Mar 16 at 12:10

pi66pi66

3,9871239

$endgroup$

This question has an open bounty worth +50
reputation from pi66 ending ending at 2019-03-25 12:10:45Z">in 5 hours.

This question has not received enough attention.

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Is $[0,1] in U$?
  $endgroup$
  – Berci
  Mar 16 at 12:30
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @Berci Yes, it is.
  $endgroup$
  – pi66
  Mar 16 at 12:53
  

  
  
  
  

add a comment | 

$begingroup$

Let $U$ be the set of all nonempty subsets of $[0,1]$ that are a union of finitely many closed intervals (where an "interval" that is a single point does not count as an interval). Does there exist a function $f:Utimes Urightarrow U$ such that for any $A,Bin U$:

(a) $f(A,B)neq [0,1]$.

(b) $f(A,B)cap Aneqemptyset$ and $f(A,B)cap Bneqemptyset$.

(c) The length (i.e. Lebesgue measure) of $f(X,B)cap A$ is maximized at $X=A$, and the length of $f(A,X)cap B$ is maximized at $X=B$.

Any two of the three conditions can be satisfied:

• $f(A,B)equiv [0,1]$ satisfy (b) and (c).




• $f(A,B)equiv Y$ for any fixed $Yneq [0,1]$ satisfy (a) and (c).




• $f(A,B)$ being any $Yneq [0,1]$ that intersects both $A$ and $B$ satisfy (a) and (b).

Satisfying all three seems to be impossible though.

(Question posted on MO five months ago but still unsolved.)

real-analysis lebesgue-measure

share|cite|improve this question

edited Mar 21 at 5:30

Eric Wofsey

190k14216348

asked Mar 16 at 12:10

pi66pi66

3,9871239

$endgroup$

share|cite|improve this question

edited Mar 21 at 5:30

Eric Wofsey

190k14216348

asked Mar 16 at 12:10

pi66pi66

3,9871239

share|cite|improve this question

edited Mar 21 at 5:30

Eric Wofsey

190k14216348

asked Mar 16 at 12:10

pi66pi66

3,9871239

edited Mar 21 at 5:30

Eric Wofsey

190k14216348

edited Mar 21 at 5:30

Eric Wofsey

190k14216348

asked Mar 16 at 12:10

pi66pi66

3,9871239

asked Mar 16 at 12:10

pi66pi66

3,9871239