Convex hulls change continuously as one point moves continuously 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.
