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Minor Math Jaxing (used $\|\cdot\|$ instead of $||\cdot||$)
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Daniele Tampieri
Daniele Tampieri
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Let $C_{lsc}(\mathbb{R}^n)$ be the space of lower semicontinuous convex functions $\mathbb{R}^n \to \mathbb{R}$. The Legendre-Fenchel (LF) transform of $f \in C_{lsc}(\mathbb{R}^n)$ is: $$ f^*(y) := \sup_{x \in \mathbb{R}^n} (\langle x, y \rangle - f(x)) $$

It is known that the LF transform is continuous and an involution on $C_{lsc}(\mathbb{R}^n)$ (Wijsman 1963). I want to know if the LF transform is Lipschitz. That is, given $f$ and $g$ lsc, is there a way to bound $||f^* - g^*||$$\|f^* - g^*\|$ by $||f-g||$$\|f-g\|$ (under a standard norm)?

I haven't found any sources that do this, so my suspicion is that the LF transform is not Lipschitz. Does anyone know if this is true, or if not a simple counterexample?

It is worth noting that Attouch and Wets ("Isometries of the Legendre-Fenchel transform") constructed norms under which the LF transform is an isometry -- however these norms are not particularly useful for me. I am looking for any results using standard norms (any of the p-norms).

Let $C_{lsc}(\mathbb{R}^n)$ be the space of lower semicontinuous convex functions $\mathbb{R}^n \to \mathbb{R}$. The Legendre-Fenchel (LF) transform of $f \in C_{lsc}(\mathbb{R}^n)$ is: $$ f^*(y) := \sup_{x \in \mathbb{R}^n} (\langle x, y \rangle - f(x)) $$

It is known that the LF transform is continuous and an involution on $C_{lsc}(\mathbb{R}^n)$ (Wijsman 1963). I want to know if the LF transform is Lipschitz. That is, given $f$ and $g$ lsc, is there a way to bound $||f^* - g^*||$ by $||f-g||$ (under a standard norm)?

I haven't found any sources that do this, so my suspicion is that the LF transform is not Lipschitz. Does anyone know if this is true, or if not a simple counterexample?

It is worth noting that Attouch and Wets ("Isometries of the Legendre-Fenchel transform") constructed norms under which the LF transform is an isometry -- however these norms are not particularly useful for me. I am looking for any results using standard norms (any of the p-norms).

Let $C_{lsc}(\mathbb{R}^n)$ be the space of lower semicontinuous convex functions $\mathbb{R}^n \to \mathbb{R}$. The Legendre-Fenchel (LF) transform of $f \in C_{lsc}(\mathbb{R}^n)$ is: $$ f^*(y) := \sup_{x \in \mathbb{R}^n} (\langle x, y \rangle - f(x)) $$

It is known that the LF transform is continuous and an involution on $C_{lsc}(\mathbb{R}^n)$ (Wijsman 1963). I want to know if the LF transform is Lipschitz. That is, given $f$ and $g$ lsc, is there a way to bound $\|f^* - g^*\|$ by $\|f-g\|$ (under a standard norm)?

I haven't found any sources that do this, so my suspicion is that the LF transform is not Lipschitz. Does anyone know if this is true, or if not a simple counterexample?

It is worth noting that Attouch and Wets ("Isometries of the Legendre-Fenchel transform") constructed norms under which the LF transform is an isometry -- however these norms are not particularly useful for me. I am looking for any results using standard norms (any of the p-norms).

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gdavtor
gdavtor
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Is the Legendre transform as an operator Lipschitz?

Let $C_{lsc}(\mathbb{R}^n)$ be the space of lower semicontinuous convex functions $\mathbb{R}^n \to \mathbb{R}$. The Legendre-Fenchel (LF) transform of $f \in C_{lsc}(\mathbb{R}^n)$ is: $$ f^*(y) := \sup_{x \in \mathbb{R}^n} (\langle x, y \rangle - f(x)) $$

It is known that the LF transform is continuous and an involution on $C_{lsc}(\mathbb{R}^n)$ (Wijsman 1963). I want to know if the LF transform is Lipschitz. That is, given $f$ and $g$ lsc, is there a way to bound $||f^* - g^*||$ by $||f-g||$ (under a standard norm)?

I haven't found any sources that do this, so my suspicion is that the LF transform is not Lipschitz. Does anyone know if this is true, or if not a simple counterexample?

It is worth noting that Attouch and Wets ("Isometries of the Legendre-Fenchel transform") constructed norms under which the LF transform is an isometry -- however these norms are not particularly useful for me. I am looking for any results using standard norms (any of the p-norms).