problem statement
For any $x \in R^d$, consider the function $$\phi(x,a) = \min_{1\leq j \leq k} \|x-a_j\|^2,$$ where $a = (a_1, \dots, a_k) \in R^{kd}$ and $\| \cdot \|$ is the $L_p$ norm for any $p \geq 1$. Let $X$ be a continuous random vector in $R^d$ with distribution $P$ such that $\int \|x\|^2 dP(x) < \infty$ and assume measurability is taken care of for all functions.
Consider for some fixed $b$ and for some $R>0$, the class of functions: $$ G_R := \{ g(x) = \phi(x,a) - \phi(x,b) : \| a - b\| \leq R \} $$
I'd like to find a function $F(x)$ and constant $R>0$ such that:
1) $F(x) \geq |g(x)|, \forall g(x) \in G_R$.
2) $\int {F(x)}^2 dP(x) = O(R^2),$ and $\forall \epsilon >0, \int {F(x)}^2 1\{F(x) > \epsilon\} dP(x) = o(R^2)$ as $R \rightarrow 0$.
my attempt
I have a solution for the case of $p=2$ (Euclidean norm) which I have trouble generalizing for other $L_p$ norms.
For $a = (a_1, \dots, a_k)$, let $A_j$ denote the set of points in $R^d$ that are closest to $a_j$ among $\{a_1, \dots, a_k \}$ according to $\| \cdot \|$).
Notice that if $x \in interior({A_j})$ and $h$ sufficiently small, \begin{equation} \label{jlinearexpansioneqn} \phi(x;{a+h}) = \|x-a_j-h_j\|^2 = \|x-a_j\|^2 - 2h_j'(x-a_j) + \|h_j\|^2. \end{equation} Hence, since we restrict analysis to continuous densities, boundaries between $A_j$ and $A_i$ have zero measure. So we have the following taylor expansion: \begin{equation} \label{linearexpansioneqn} \phi(x;{a+h}) = \phi(x;{a}) + {h}'\delta(x,{a}) + \|{{h}}\|r(x,{a},{h}), \end{equation} where $\delta(x,{a})$ is the gradient of $\phi(x;{a})$, the $k$ vector of $L^2(P)$ functions with element $j$ equal to $-2(x-a_j) 1\{x\in A_j\}$.
Furthermore,
$$ r(x,{a},{h}) = \sum_{j=1}^k \frac{\|{h_j}\|^2 1\{ x \in A_j\} } {{\|{h}\|}} \rightarrow 0 \text{ as } {h} \rightarrow 0 \quad \text{ a.s. in } x .$$
In addition, \begin{align} \label{remainderdominationeqn} \begin{split} |r(x,{a},{h})| &\leq {\|{h}\|}^{-1} ( |{h}'\delta(x,{a})| + \max_j | \|{x-a_j-h_j}\|^2 - \|{x-a_j}\|^2 | ) \\ &\leq \|{\delta(x,{a})}\| +{\|{h}\|}^{-1} \sum_j | \|{x-a_j-h_j}\|^2 - \|{x-a_j}\|^2 | ) \\ &\leq C(1+\|{x}\|) \end{split} \end{align} for some $C>0$ and ${h}$ sufficiently small.
Let $R>0$ be small enough such that the above domination is true for $a=b,h=a-b$. Then by the expansion above and Cauchy-Schwarz: \begin{align} \begin{split} |g(x) | & = \left| \phi(x;{a}) - \phi(x;b) \right| \\ &= \left| ({a} - b)'\delta(x,b) + { \|{a} - b}\| r(x,b,{a} - b) \right| \\ &\leq {\|{a} - b}\| \|{ \delta(x,b)}\| + C { \|{a} - b}\|(1+\|{x}\|) \\ &= B(x) \|{{a} - b}\|, \end{split} \end{align} where $B(x) = \|{ \delta(x,b)}\| + C(1+\|{x}\|) \in L^2 (P)$. The dominating function $F(x) = RB(x)$ satisfies 1) and 2).
Thanks for your help!