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YCor
YCor
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Reference request: Smoothness of Minkowski Functionalfunctional is equivalent to smoothness of boundary

Let $C\subseteq \mathbb{R}^n$ be a convex body containing $0$ in its interior. I recently read that Minkowski functional of $C$, $$ f_C(x):=\inf\{t>0:\frac1{t}\cdot x\in C\}, $$$$ f_C(x):=\inf\Big\{t>0:\frac1{t}\cdot x\in C\Big\}, $$ is $C^1$ if and only if $C$ has a $C^1$-boundary. However, I can't find a reference for this; would someone happen to know one?

Reference request: Smoothness of Minkowski Functional is equivalent to smoothness of boundary

Let $C\subseteq \mathbb{R}^n$ be a convex body containing $0$ in its interior. I recently read that Minkowski functional of $C$, $$ f_C(x):=\inf\{t>0:\frac1{t}\cdot x\in C\}, $$ is $C^1$ if and only if $C$ has a $C^1$-boundary. However, I can't find a reference for this; would someone happen to know one?

Smoothness of Minkowski functional is equivalent to smoothness of boundary

Let $C\subseteq \mathbb{R}^n$ be a convex body containing $0$ in its interior. I recently read that Minkowski functional of $C$, $$ f_C(x):=\inf\Big\{t>0:\frac1{t}\cdot x\in C\Big\}, $$ is $C^1$ if and only if $C$ has a $C^1$-boundary. However, I can't find a reference for this; would someone happen to know one?

Corrected misspelled name.
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Margaret Friedland
Margaret Friedland
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Reference request: Smoothness of MinkowksiMinkowski Functional is equivalent to smoothness of boundary

Let $C\subseteq \mathbb{R}^n$ be a convex body containing $0$ in its interior. I recently read that Minkowski functional of $C$' Minkowksi functional, $$ f_C(x):=\inf\{t>0:\frac1{t}\cdot x\in C\}, $$ is $C^1$ if and only if $C$ has a $C^1$-boundary. However, I can't find a reference for this; would someone happen to know one?

Reference request: Smoothness of Minkowksi Functional is equivalent to smoothness of boundary

Let $C\subseteq \mathbb{R}^n$ be a convex body containing $0$ in its interior. I recently read that $C$' Minkowksi functional $$ f_C(x):=\inf\{t>0:\frac1{t}\cdot x\in C\}, $$ is $C^1$ if and only if $C$ has a $C^1$-boundary. However, I can't find a reference for this; would someone happen to know one?

Reference request: Smoothness of Minkowski Functional is equivalent to smoothness of boundary

Let $C\subseteq \mathbb{R}^n$ be a convex body containing $0$ in its interior. I recently read that Minkowski functional of $C$, $$ f_C(x):=\inf\{t>0:\frac1{t}\cdot x\in C\}, $$ is $C^1$ if and only if $C$ has a $C^1$-boundary. However, I can't find a reference for this; would someone happen to know one?

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ABIM
ABIM
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Reference request: Smoothness of Minkowksi Functional is equivalent to smoothness of boundary

Let $C\subseteq \mathbb{R}^n$ be a convex body containing $0$ in its interior. I recently read that $C$' Minkowksi functional $$ f_C(x):=\inf\{t>0:\frac1{t}\cdot x\in C\}, $$ is $C^1$ if and only if $C$ has a $C^1$-boundary. However, I can't find a reference for this; would someone happen to know one?