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I agree with @Mahdi: For a function $g \colon H \to (- \infty, \infty]$, where $H$ is a Hilbert space (this even makes sense in a reflexive Banach space, but one has to replace $H$ by $H^*$ sometimes in the sequel), $g^{**}(x) = \sup_{y \in H} \langle x, y \rangle - g^*(y)$ is the biconjugate of $g$ (where $g^*$ is the convex conjugate of $g$), which is convex and lower semicontinuous, because it is the supremum of the collection of affine (and hence convex and continuous) functions $f_{y}(x) := \langle x, y \rangle - g^*(y)$ for $y \in H$ with $g^*(y) < \infty$. We have $g^{**} \le g$ with equality if and only if $g$ is proper, convex and lower semicontinuous. If $\text{dom}(g^*) := \{ x \in H: g^*(x) \in \mathbb R \} \ne \emptyset$, then $g^{**}$ is the largest convex and lower semicontinuous function minorizing $g$ (Prop. 9.8(i) together with Prop. 13.39 in Bauschke and Combettes' book Convex Analysis and Monotone Operators in Hilbert spaces in chapter 13.

Since this has nothing to do with $H = \mathbb R^d$, the Lipschitzian properties of convex functions in $\mathbb R^d$ are a red herring.

I agree with @Mahdi: For a proper function $g \colon H \to (- \infty, \infty]$, where $H$ is a Hilbert space (this even makes sense in a reflexive Banach space, but one has to replace $H$ by $H^*$ sometimes in the sequel), $g^{**}(x) = \sup_{y \in H} \langle x, y \rangle - g^*(y)$ is the biconjugate of $g$ (where $g^*$ is the convex conjugate of $g$), which is convex and lower semicontinuous, because it is the supremum of the collection of affine (and hence convex and continuous) functions $f_{y}(x) := \langle x, y \rangle - g^*(y)$ for $y \in H$ with $g^*(y) < \infty$. We have $g^{**} \le g$ with equality if and only if $g$ is proper, convex and lower semicontinuous. If $\text{dom}(g^*) := \{ x \in H: g^*(x) \in \mathbb R \} \ne \emptyset$, then $g^{**}$ is the largest convex and lower semicontinuous function minorizing $g$ (Prop. 9.8(i) together with Prop. 13.39 in Bauschke and Combettes' book Convex Analysis and Monotone Operators in Hilbert spaces in chapter 13.

Since this has nothing to do with $H = \mathbb R^d$, the Lipschitzian properties of convex functions in $\mathbb R^d$ are a red herring.

I agree with @Mahdi: For a function $g \colon H \to (- \infty, \infty]$, where $H$ is a Hilbert space (or even a reflexive Banach space), $g^{**}(x) = \sup_{y \in H} \langle x, y \rangle - g^*(y)$ is the biconjugate of $g$ (where $g^*$ is the convex conjugate of $g$), which is convex and lower semicontinuous, because it is the supremum of the collection of affine (and hence convex and continuous) functions $f_{y}(x) := \langle x, y \rangle - g^*(y)$ for $y \in H$ with $g^*(y) < \infty$. We have $g^{**} \le g$ with equality if and only if $g$ is proper, convex and lower semicontinuous and $g^{**}$ is the largest convex and lower semicontinuous function minorizing $g$ (Prop. 9.8(i) together with Prop. 13.38). You can read more about this in Bauschke and Combettes' book Convex Analysis and Monotone Operators in Hilbert spaces in chapter 13.

Since this has nothing to do with $H = \mathbb R^d$, the Lipschitzian properties of convex functions in $\mathbb R^d$ are a red herring.

I agree with @Mahdi: For a function $g \colon H \to (- \infty, \infty]$, where $H$ is a Hilbert space (this even makes sense in a reflexive Banach space, but one has to replace $H$ by $H^*$ sometimes in the sequel), $g^{**}(x) = \sup_{y \in H} \langle x, y \rangle - g^*(y)$ is the biconjugate of $g$ (where $g^*$ is the convex conjugate of $g$), which is convex and lower semicontinuous, because it is the supremum of the collection of affine (and hence convex and continuous) functions $f_{y}(x) := \langle x, y \rangle - g^*(y)$ for $y \in H$ with $g^*(y) < \infty$. We have $g^{**} \le g$ with equality if and only if $g$ is proper, convex and lower semicontinuous. If $\text{dom}(g^*) := \{ x \in H: g^*(x) \in \mathbb R \} \ne \emptyset$, then $g^{**}$ is the largest convex and lower semicontinuous function minorizing $g$ (Prop. 9.8(i) together with Prop. 13.39 in Bauschke and Combettes' book Convex Analysis and Monotone Operators in Hilbert spaces in chapter 13.

Since this has nothing to do with $H = \mathbb R^d$, the Lipschitzian properties of convex functions in $\mathbb R^d$ are a red herring.

I agree with @Mahdi: For a function $g \colon H \to (- \infty, \infty]$, where $H$ is a Hilbert space (or even a reflexive Banach space), $g^{**}(x) = \sup_{y \in H} \langle x, y \rangle - g^*(y)$ is the biconjugate of $g$ (where $g^*$ is the convex conjugate of $g$), which is convex and lower semicontinuous, because it is the supremum of the affine (and hence convex and continuous) the collection of functions $f_{y}(x) := \langle x, y \rangle - g^*(y)$ for $y \in H$ with $g^*(y) < \infty$. We have $g^{**} \le g$ with equality if and only if $g$ is proper, convex and lower semicontinuous and $g^{**}$ is the largest convex and lower semicontinuous function minorizing $g$ (Prop. 9.8(i) together with Prop. 13.38). You can read more about this in Bauschke and Combettes' book Convex Analysis and Monotone Operators in Hilbert spaces in chapter 13.

Since this has nothing to do with $H = \mathbb R^d$, the Lipschitzian properties of convex functions in $\mathbb R^d$ are a red herring.

I agree with @Mahdi: For a function $g \colon H \to (- \infty, \infty]$, where $H$ is a Hilbert space (or even a reflexive Banach space), $g^{**}(x) = \sup_{y \in H} \langle x, y \rangle - g^*(y)$ is the biconjugate of $g$ (where $g^*$ is the convex conjugate of $g$), which is convex and lower semicontinuous, because it is the supremum of the collection of affine (and hence convex and continuous) functions $f_{y}(x) := \langle x, y \rangle - g^*(y)$ for $y \in H$ with $g^*(y) < \infty$. We have $g^{**} \le g$ with equality if and only if $g$ is proper, convex and lower semicontinuous and $g^{**}$ is the largest convex and lower semicontinuous function minorizing $g$ (Prop. 9.8(i) together with Prop. 13.38). You can read more about this in Bauschke and Combettes' book Convex Analysis and Monotone Operators in Hilbert spaces in chapter 13.

Since this has nothing to do with $H = \mathbb R^d$, the Lipschitzian properties of convex functions in $\mathbb R^d$ are a red herring.