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Ayanna Trujillo

Ayanna Trujillo

Answered question

2022-06-15

Let γ : [ 0 , 1 ] S 2 be a smooth curve. Consider the following set
t [ 0 , 1 ] s p a n ( γ ( t ) ) R 3
My question is: does this set have Hausdorff dimension at most 2, and as a result it has zero 3-dimensional Lebesgue measure?. Note that if the smoothness condition is removed then the answer is trivially false.

Answer & Explanation

Anika Stevenson

Anika Stevenson

Beginner2022-06-16Added 19 answers

It seems that the answer is pretty easy......
The set
t [ 0 , 1 ] s p a n ( γ ( t ) )
is just the graph of the smooth mapping Γ ( x , t ) := x γ ( t ) over R × [ 0 , 1 ]. Therefore its Hausdorff dimension is exactly 2.

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