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L.Ambrosio, in paper [1] writes:

Let $g:\mathbb{R}\times\mathbb{R}^n\rightarrow\mathbb{R}$ be a function (...)

 

for every $s\in\mathbb{R}$, $z\in\mathbb{R}^n$; we denote, with a slight abuse of notation, by $g^{**}$ the function defined by

 

$$g^{**}(s,z) = [g(s,\cdot)]^{**}(z)$$

 

(...) where $^{**}$ denotes the lower semicontinous and convex envelope.

My question is: What exactly does "lower semicontinuous and convex envelope" mean? If you calculate the convex envelope of $g(s,\cdot)$ you end up with a function $C(g)(s,\cdot):\mathbb{R}^n\rightarrow\mathbb{R}$ which is convex, and therefore lower semicontinuous.

[1] Ambrosio, Luigi, Relaxation of autonomous functionals with discontinuous integrands, Ann. Univ. Ferrara, Nuova Ser., Sez. VII 34, 21-47 (1988). ZBL0691.49011.

L.Ambrosio, in paper [1] writes:

Let $g:\mathbb{R}\times\mathbb{R}^n\rightarrow\mathbb{R}$ be a function (...)

 

for every $s\in\mathbb{R}$, $z\in\mathbb{R}^n$; we denote, with a slight abuse of notation, by $g^{**}$ the function defined by

 

$$g^{**}(s,z) = [g(s,\cdot)]^{**}(z)$$

 

(...) where $^{**}$ denotes the lower semicontinous and convex envelope.

My question is: What exactly does "lower semicontinuous and convex envelope" mean? If you calculate the convex envelope of $g(s,\cdot)$ you end up with a function $C(g)(s,\cdot):\mathbb{R}^n\rightarrow\mathbb{R}$ which is convex, and therefore lower semicontinuous.

[1] Ambrosio, Luigi, Relaxation of autonomous functionals with discontinuous integrands, Ann. Univ. Ferrara, Nuova Ser., Sez. VII 34, 21-47 (1988). ZBL0691.49011.

L.Ambrosio, in paper [1] writes:

Let $g:\mathbb{R}\times\mathbb{R}^n\rightarrow\mathbb{R}$ be a function (...)

for every $s\in\mathbb{R}$, $z\in\mathbb{R}^n$; we denote, with a slight abuse of notation, by $g^{**}$ the function defined by

$$g^{**}(s,z) = [g(s,\cdot)]^{**}(z)$$

(...) where $^{**}$ denotes the lower semicontinous and convex envelope.

My question is: What exactly does "lower semicontinuous and convex envelope" mean? If you calculate the convex envelope of $g(s,\cdot)$ you end up with a function $C(g)(s,\cdot):\mathbb{R}^n\rightarrow\mathbb{R}$ which is convex, and therefore lower semicontinuous.

[1] Ambrosio, Luigi, Relaxation of autonomous functionals with discontinuous integrands, Ann. Univ. Ferrara, Nuova Ser., Sez. VII 34, 21-47 (1988). ZBL0691.49011.

A grammar error (Used "¿" to begin a question).
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L.Ambrosio, in paper [1] writes:

Let $g:\mathbb{R}\times\mathbb{R}^n\rightarrow\mathbb{R}$ be a function (...)

for every $s\in\mathbb{R}$, $z\in\mathbb{R}^n$; we denote, with a slight abuse of notation, by $g^{**}$ the function defined by

$$g^{**}(s,z) = [g(s,\cdot)]^{**}(z)$$

(...) where $^{**}$ denotes the lower semicontinous and convex envelope.

My question is: ¿WhatWhat exactly does "lower semicontinuous and convex"convex envelope" mean? If you calculate the convex envelope of $g(s,\cdot)$ you end up with a function $C(g)(s,\cdot):\mathbb{R}^n\rightarrow\mathbb{R}$ which is convex, and therefore lower semicontinuous.

[1] Ambrosio, Luigi, Relaxation of autonomous functionals with discontinuous integrands, Ann. Univ. Ferrara, Nuova Ser., Sez. VII 34, 21-47 (1988). ZBL0691.49011.

L.Ambrosio, in paper [1] writes:

Let $g:\mathbb{R}\times\mathbb{R}^n\rightarrow\mathbb{R}$ be a function (...)

for every $s\in\mathbb{R}$, $z\in\mathbb{R}^n$; we denote, with a slight abuse of notation, by $g^{**}$ the function defined by

$$g^{**}(s,z) = [g(s,\cdot)]^{**}(z)$$

(...) where $^{**}$ denotes the lower semicontinous and convex envelope.

My question is: ¿What exactly does "lower semicontinuous and convex" mean? If you calculate the convex envelope of $g(s,\cdot)$ you end up with a function $C(g)(s,\cdot):\mathbb{R}^n\rightarrow\mathbb{R}$ which is convex, and therefore lower semicontinuous.

[1] Ambrosio, Luigi, Relaxation of autonomous functionals with discontinuous integrands, Ann. Univ. Ferrara, Nuova Ser., Sez. VII 34, 21-47 (1988). ZBL0691.49011.

L.Ambrosio, in paper [1] writes:

Let $g:\mathbb{R}\times\mathbb{R}^n\rightarrow\mathbb{R}$ be a function (...)

for every $s\in\mathbb{R}$, $z\in\mathbb{R}^n$; we denote, with a slight abuse of notation, by $g^{**}$ the function defined by

$$g^{**}(s,z) = [g(s,\cdot)]^{**}(z)$$

(...) where $^{**}$ denotes the lower semicontinous and convex envelope.

My question is: What exactly does "lower semicontinuous and convex envelope" mean? If you calculate the convex envelope of $g(s,\cdot)$ you end up with a function $C(g)(s,\cdot):\mathbb{R}^n\rightarrow\mathbb{R}$ which is convex, and therefore lower semicontinuous.

[1] Ambrosio, Luigi, Relaxation of autonomous functionals with discontinuous integrands, Ann. Univ. Ferrara, Nuova Ser., Sez. VII 34, 21-47 (1988). ZBL0691.49011.

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vicubso
vicubso
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Lower semicontinuous and convex envelope

L.Ambrosio, in paper [1] writes:

Let $g:\mathbb{R}\times\mathbb{R}^n\rightarrow\mathbb{R}$ be a function (...)

for every $s\in\mathbb{R}$, $z\in\mathbb{R}^n$; we denote, with a slight abuse of notation, by $g^{**}$ the function defined by

$$g^{**}(s,z) = [g(s,\cdot)]^{**}(z)$$

(...) where $^{**}$ denotes the lower semicontinous and convex envelope.

My question is: ¿What exactly does "lower semicontinuous and convex" mean? If you calculate the convex envelope of $g(s,\cdot)$ you end up with a function $C(g)(s,\cdot):\mathbb{R}^n\rightarrow\mathbb{R}$ which is convex, and therefore lower semicontinuous.

[1] Ambrosio, Luigi, Relaxation of autonomous functionals with discontinuous integrands, Ann. Univ. Ferrara, Nuova Ser., Sez. VII 34, 21-47 (1988). ZBL0691.49011.