I am seeking a reference or a succinct proof of this claim: Let $H$ be the convex hull of a finite set of points $p_1,p_2,\ldots$ in $\mathbb{R}^d$. Move one point $p_1$ along a continuous rectifiable path $\pi$. Then $H$ changes continuously under the Hausdorff distance. A small move of $p_1$ along $\pi$ results in a small change in the hulls viewed as sets of points.
In my situation, I have convex bodies instead of the points $p_i$, but I think that does not matter. I would be happy with a proof for points. Thanks!
We have the following definition for the convex hull of $n$ points: $$ H = \left\{\left. \sum_{i=1}^{n}\alpha_ip_i \right| 0\le \alpha_i\le 1 \text{ and } \sum_{i=1}^{n}\alpha_i=1\right\}. $$ Let $H(\delta)$ be the convex hull that we obtain when we move $p_1$ along $\pi$ for a distance $\delta$. Then for any $x = \sum_{i=1}^{n}\alpha_ip_i\in H$ we have $$ \inf_{y\in Y}d(x,y) \le d\left(\sum_{i=1}^{n}\alpha_ip_i,\alpha_1(p_1+\delta) + \sum_{i=2}^{n}\alpha_ip_i\right) = d(0,\alpha_1\delta) = \alpha_1\delta \le \delta. $$ Similarly you can show that we have for any $y\in Y$ that $\inf_{x\in X}d(x,y) \le \delta$. We conclude that $$ d(H,H(\delta)) = \max\left\{\sup_{x\in X}\inf_{y\in Y}d(x,y) ,\sup_{y\in Y}\inf_{x\in X}d(x,y) \right\} \le \delta. $$ Thus we have a continuous function.