Mixing
Boundary of convex subset diffeomorphic to the sphere
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Suppose you have a strictly convex set $Omega subset mathbb{R}^n$, $n geq 0$, bounded, with non-empty interior. Without loss of generality I will assume $0 in text{Int} Omega$. Thus there exists a clear homeomorphism $x to frac{x}{|x|}$ between the boundary of $Omega$ and the sphere.
Seems to me that this map is actually a $C^1$-diffeomorphism, is it true without further assumption ? What if we also assume the boundary of $Omega$ to be $C^1$ ?
differential-geometry convex-analysis
asked Jan 10 at 10:34
Gâteau-GalloisGâteau-Gallois
362112
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add a comment |
$begingroup$
Suppose you have a strictly convex set $Omega subset mathbb{R}^n$, $n geq 0$, bounded, with non-empty interior. Without loss of generality I will assume $0 in text{Int} Omega$. Thus there exists a clear homeomorphism $x to frac{x}{|x|}$ between the boundary of $Omega$ and the sphere.
Seems to me that this map is actually a $C^1$-diffeomorphism, is it true without further assumption ? What if we also assume the boundary of $Omega$ to be $C^1$ ?
differential-geometry convex-analysis
asked Jan 10 at 10:34
Gâteau-GalloisGâteau-Gallois
362112
$endgroup$
add a comment |
$begingroup$
Suppose you have a strictly convex set $Omega subset mathbb{R}^n$, $n geq 0$, bounded, with non-empty interior. Without loss of generality I will assume $0 in text{Int} Omega$. Thus there exists a clear homeomorphism $x to frac{x}{|x|}$ between the boundary of $Omega$ and the sphere.
Seems to me that this map is actually a $C^1$-diffeomorphism, is it true without further assumption ? What if we also assume the boundary of $Omega$ to be $C^1$ ?
differential-geometry convex-analysis
asked Jan 10 at 10:34
Gâteau-GalloisGâteau-Gallois
362112
$endgroup$
Suppose you have a strictly convex set $Omega subset mathbb{R}^n$, $n geq 0$, bounded, with non-empty interior. Without loss of generality I will assume $0 in text{Int} Omega$. Thus there exists a clear homeomorphism $x to frac{x}{|x|}$ between the boundary of $Omega$ and the sphere.
Seems to me that this map is actually a $C^1$-diffeomorphism, is it true without further assumption ? What if we also assume the boundary of $Omega$ to be $C^1$ ?
differential-geometry convex-analysis
differential-geometry convex-analysis
asked Jan 10 at 10:34
Gâteau-GalloisGâteau-Gallois
362112
asked Jan 10 at 10:34
Gâteau-GalloisGâteau-Gallois
362112
asked Jan 10 at 10:34
Gâteau-GalloisGâteau-Gallois
362112
asked Jan 10 at 10:34
Gâteau-GalloisGâteau-Gallois
362112
asked Jan 10 at 10:34
Gâteau-GalloisGâteau-Gallois
362112
362112
add a comment |
add a comment |
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