Consider a one-parameter family of convex bodies $\Sigma_\varepsilon$ in $\mathbb{R}^n$, $-1 < \varepsilon < 1$. Let $z$ be a point not contained in $\Sigma_0$, and let $\pi^{\Sigma_\varepsilon}(z)$ be the unique point in $\Sigma_\varepsilon$ closest to $z$. How does $\pi^{\Sigma_\varepsilon}(z)$ change with respect to $\varepsilon$? I'm mostly interested in $d\pi^{\Sigma_\varepsilon}(z)/d\varepsilon |_{\varepsilon = 0}$, if the derivative exists.
I have not specified how $\Sigma_\varepsilon$ is parametrized. To make the question precise, assume $\Sigma_\varepsilon = \{x | f(\varepsilon, x) \leq 0\}$, where each $f(\varepsilon, \cdot)$ is a convex function, and the dependence on $\varepsilon$ is smooth. If other forms of parametrization give more meaningful results, please suggest.
My thoughts: If $f(\varepsilon, x)$ is sufficiently smooth, then $z - \pi^{\Sigma_\varepsilon}(z)$ should be parallel to $\nabla_x f$. This leads to a family of equations involving $x, \varepsilon$, and I could perform implicit differentiation.
