Let $K\subseteq\mathbb R^d$ be a compact convex body with non-empty interior, and $E$ be a $(d-1)$-dimensional linear subspace of $\mathbb R^d$. Let $\theta\in\mathbb R^d$ be the unit vector such that $\langle x,\theta\rangle = 0$ for all $x\in E$.

The Minkowski-Stein formula states that $$ \mu_{d-1}(K\cap E) = \lim_{\varepsilon\to 0^+}2\varepsilon^{-1}\mu_d\left(\{x\in K: |\langle x,\theta\rangle|\leq\varepsilon\}\right), $$ where $\mu_{d-1}$ and $\mu_d$ denote the Lebesgue measure on $\mathbb R^{d-1}$ and $\mathbb R^d$.

My question is:

Suppose $K$ or $K\cap E$ satisfy certain regularity/smoothness conditions, is it possible to control the error between $2\varepsilon^{-1}\mu_d(\{x\in K:|\langle x,\theta\rangle|\leq\varepsilon\})$ and $\mu_{d-1}(K\cap E)$ using quantities involving $K$ and $\varepsilon$? As special cases, what if $K$ is the unit $\ell_p$ ball $\{|x_1|^p+\cdots+|x_d|^p\leq 1\}$ where $p\in(1,\infty)$?