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Generalization of Kakeya problem
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Say a needle of unit length is on a flat surface and its head is pointing to the right, we apply a motion to the pen so that it's head ends up pointing to the left. For example : if we rotate the pen around its middle then we swept a surface that is a disk of radius 1/2. And the area is Ã€/4
We can find motions with area smaller than that and the smallest possible area has infimmum equal to 0 thanks to Kakeya.
I would like to hear some ideas for the case when the needle is not a straight line but any curve of length 1, what is the smallest area that we could possibly sweep ?
calculus geometry
edited Jul 23 at 19:25
Bernard
110k635103
asked Jul 23 at 19:12
hra
1,3011317
add a comment |Â
up vote
0
down vote
favorite
Say a needle of unit length is on a flat surface and its head is pointing to the right, we apply a motion to the pen so that it's head ends up pointing to the left. For example : if we rotate the pen around its middle then we swept a surface that is a disk of radius 1/2. And the area is Ã€/4
We can find motions with area smaller than that and the smallest possible area has infimmum equal to 0 thanks to Kakeya.
I would like to hear some ideas for the case when the needle is not a straight line but any curve of length 1, what is the smallest area that we could possibly sweep ?
calculus geometry
edited Jul 23 at 19:25
Bernard
110k635103
asked Jul 23 at 19:12
hra
1,3011317
If the needle with length $1$ is wrapped into a curve with radius $0$ and $infty$ winds from head to tail, the area swept will be $0$. Perhaps somewhere close to this condition?
– Weather Vane
Jul 23 at 19:23
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Say a needle of unit length is on a flat surface and its head is pointing to the right, we apply a motion to the pen so that it's head ends up pointing to the left. For example : if we rotate the pen around its middle then we swept a surface that is a disk of radius 1/2. And the area is Ã€/4
We can find motions with area smaller than that and the smallest possible area has infimmum equal to 0 thanks to Kakeya.
I would like to hear some ideas for the case when the needle is not a straight line but any curve of length 1, what is the smallest area that we could possibly sweep ?
calculus geometry
edited Jul 23 at 19:25
Bernard
110k635103
asked Jul 23 at 19:12
hra
1,3011317
Say a needle of unit length is on a flat surface and its head is pointing to the right, we apply a motion to the pen so that it's head ends up pointing to the left. For example : if we rotate the pen around its middle then we swept a surface that is a disk of radius 1/2. And the area is Ã€/4
We can find motions with area smaller than that and the smallest possible area has infimmum equal to 0 thanks to Kakeya.
I would like to hear some ideas for the case when the needle is not a straight line but any curve of length 1, what is the smallest area that we could possibly sweep ?
calculus geometry
edited Jul 23 at 19:25
Bernard
110k635103
asked Jul 23 at 19:12
hra
1,3011317
edited Jul 23 at 19:25
Bernard
110k635103
edited Jul 23 at 19:25
Bernard
110k635103
edited Jul 23 at 19:25
Bernard
110k635103
110k635103
asked Jul 23 at 19:12
hra
1,3011317
asked Jul 23 at 19:12
hra
1,3011317
asked Jul 23 at 19:12
hra
1,3011317
1,3011317
If the needle with length $1$ is wrapped into a curve with radius $0$ and $infty$ winds from head to tail, the area swept will be $0$. Perhaps somewhere close to this condition?
– Weather Vane
Jul 23 at 19:23
add a comment |Â
If the needle with length $1$ is wrapped into a curve with radius $0$ and $infty$ winds from head to tail, the area swept will be $0$. Perhaps somewhere close to this condition?
– Weather Vane
Jul 23 at 19:23
If the needle with length $1$ is wrapped into a curve with radius $0$ and $infty$ winds from head to tail, the area swept will be $0$. Perhaps somewhere close to this condition?
– Weather Vane
Jul 23 at 19:23
If the needle with length $1$ is wrapped into a curve with radius $0$ and $infty$ winds from head to tail, the area swept will be $0$. Perhaps somewhere close to this condition?
– Weather Vane
Jul 23 at 19:23
add a comment |Â
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If the needle with length $1$ is wrapped into a curve with radius $0$ and $infty$ winds from head to tail, the area swept will be $0$. Perhaps somewhere close to this condition?
– Weather Vane
Jul 23 at 19:23