What is a tube in $mathbb{R}^n$?











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Lebesgue's differentiation theorem states that if $x$ is a point in $mathbb{R}^n$ and $f:mathbb{R}^nrightarrowmathbb{R}$ is a Lebesgue integrable function, then the limit of $frac{int_B f dlambda}{lambda(B)}$ over all balls $B$ centered at $x$ as the diameter of $B$ goes to $0$ is equal almost everywhere to $f(x)$. But if you replace balls with other kinds of set with diameter going to $0$, this need not be true. For instance it need not be true if you replace balls with rectangles.



But I just came across a journal paper which shows that if you take the collection of all "tubes" in $mathbb{R}^n$ oriented in certain directions, then the Lebesgue differentiation holds true for this collection for $L^p$ functions with $p>1$. But my question is, what exactly is a tube in $mathbb{R}^n$ as the term is used in this paper? The paper doesn't provide any definition as far as I can tell.



Is it like a cylinder, or what?






geometry measure-theory definition lebesgue-integral lebesgue-measure







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    Lebesgue's differentiation theorem states that if $x$ is a point in $mathbb{R}^n$ and $f:mathbb{R}^nrightarrowmathbb{R}$ is a Lebesgue integrable function, then the limit of $frac{int_B f dlambda}{lambda(B)}$ over all balls $B$ centered at $x$ as the diameter of $B$ goes to $0$ is equal almost everywhere to $f(x)$. But if you replace balls with other kinds of set with diameter going to $0$, this need not be true. For instance it need not be true if you replace balls with rectangles.



    But I just came across a journal paper which shows that if you take the collection of all "tubes" in $mathbb{R}^n$ oriented in certain directions, then the Lebesgue differentiation holds true for this collection for $L^p$ functions with $p>1$. But my question is, what exactly is a tube in $mathbb{R}^n$ as the term is used in this paper? The paper doesn't provide any definition as far as I can tell.



    Is it like a cylinder, or what?






    geometry measure-theory definition lebesgue-integral lebesgue-measure







    share|cite|improve this question
























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      up vote
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      Lebesgue's differentiation theorem states that if $x$ is a point in $mathbb{R}^n$ and $f:mathbb{R}^nrightarrowmathbb{R}$ is a Lebesgue integrable function, then the limit of $frac{int_B f dlambda}{lambda(B)}$ over all balls $B$ centered at $x$ as the diameter of $B$ goes to $0$ is equal almost everywhere to $f(x)$. But if you replace balls with other kinds of set with diameter going to $0$, this need not be true. For instance it need not be true if you replace balls with rectangles.



      But I just came across a journal paper which shows that if you take the collection of all "tubes" in $mathbb{R}^n$ oriented in certain directions, then the Lebesgue differentiation holds true for this collection for $L^p$ functions with $p>1$. But my question is, what exactly is a tube in $mathbb{R}^n$ as the term is used in this paper? The paper doesn't provide any definition as far as I can tell.



      Is it like a cylinder, or what?






      geometry measure-theory definition lebesgue-integral lebesgue-measure







      share|cite|improve this question













      Lebesgue's differentiation theorem states that if $x$ is a point in $mathbb{R}^n$ and $f:mathbb{R}^nrightarrowmathbb{R}$ is a Lebesgue integrable function, then the limit of $frac{int_B f dlambda}{lambda(B)}$ over all balls $B$ centered at $x$ as the diameter of $B$ goes to $0$ is equal almost everywhere to $f(x)$. But if you replace balls with other kinds of set with diameter going to $0$, this need not be true. For instance it need not be true if you replace balls with rectangles.



      But I just came across a journal paper which shows that if you take the collection of all "tubes" in $mathbb{R}^n$ oriented in certain directions, then the Lebesgue differentiation holds true for this collection for $L^p$ functions with $p>1$. But my question is, what exactly is a tube in $mathbb{R}^n$ as the term is used in this paper? The paper doesn't provide any definition as far as I can tell.



      Is it like a cylinder, or what?






      geometry measure-theory definition lebesgue-integral lebesgue-measure




      geometry measure-theory definition lebesgue-integral lebesgue-measure






      share|cite|improve this question













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      asked 8 hours ago









      Keshav Srinivasan

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