Bounded bowl-shaped losses have Gaussian-centered Lipschitz minorants

Each sublevel $A_k=\{L\le k\delta \}$ is convex and symmetric by hypothesis. Its closure $C_k$ is closed; it is convex by [F10] when $A_k$ is nonempty, and is convex by vacuity when $A_k=\mathrm{\varnothing }$. If $x\in C_k$, [F11] supplies $x_j\in A_k$ with $x_j\to x$; then $-x_j\in A_k$ and $-x\in C_k$, so $C_k$ is symmetric (the empty case is symmetric as well). For a nonempty closed convex $C$, choose $c_x,c_y\in C$ within $ϵ$ of $d(x,C),d(y,C)$ using [F6]. Their convex combination lies in $C$, and [F7] gives $d((1-t)x+ty,C)\le (1-t)d(x,C)+td(y,C)+ϵ$. Let $ϵ\downarrow 0$ to prove convexity of the distance function. When $C=-C$, replacing each $c$ by $-c$ in the infimum shows $d(-x,C)=d(x,C)$. Thus for a nonempty proper $C_k$, $q(x)=\min (1,md(x,C_k))$ is symmetric; its sublevel at $0\le t<1$ is $\{d(x,C_k)\le t\mathrm{/}m\}$, which is convex, and its sublevel for $t\ge 1$ is the whole space. It is Borel and $m$-Lipschitz by [F1] and [F9]. For empty $C_k$ or $C_k={\mathbb{R}}^d$, the stated constant conventions have the same properties.

If $L(x)\le k\delta$, then $x\in C_k$ and the $k$th term is zero. Every remaining term is at most one, so $0\le g_{N,m}(x)\le \delta \mathrm{\#}\{1\le k<N:k\delta <L(x)\}\le L(x)$. The finite sum is bounded by $M$ and has Lipschitz constant at most $Mm$. Apply F3 separately to each summand. Every summand's expected loss is minimized at $a=b$ (use its symmetry to replace $b-H$ by $H-b$), so the same action minimizes their sum by [F8]. This proves the first displayed identity without assuming the sum quasi-convex.

Since $C_k$ is closed, its $k$th term increases to ${\mathbf{1}}_{C_k^c}$ as $m\to \mathrm{\infty }$. If $C_k={\mathbb{R}}^d$, then [F10] gives $\mathrm{int}\{L\le k\delta \}={\mathbb{R}}^d$, so the level set is all of ${\mathbb{R}}^d$ and both indicators vanish. If $C_k=\mathrm{\varnothing }$, the level set is empty and both indicators equal one. Otherwise the difference between $\{L>k\delta \}$ and $C_k^c$ is contained in the boundary of the convex set $\{L\le k\delta \}$, which is null by [F2]. To justify the Gaussian nullity, [F4] realizes $H-b=AG$. By [F12], repeated integration of the scalar densities gives $P(G\in E)={\int }_E{\prod }_{j=1}^d\varphi (z_j)d{\lambda }_d(z)$ for every Borel $E$ (for $d=1$ this is the scalar definition). The closed boundary $D$ is Borel and null, and $A^{-1}D$ is null by [F13], so $P(H-b\in D)=P(G\in A^{-1}D)=0$. Therefore F5 gives ${\lim }_{m}E{g}_{N,m}(H-b)=E[\delta {\sum }_{k=1}^{N-1}{\mathbf{1}}_{\{L(H-b)>k\delta \}}]$. The expression inside the expectation differs from $L(H-b)$ by at most $\delta$: count the grid thresholds strictly below each value in $[0,M]$. Let $N\to \mathrm{\infty }$ to obtain the second identity.