Show that is $K,L$ are convex bodies then $h_{\text{conv}(K\cup L)}=\max\{h_K,h_L\}$

Question

Show that if $K$ and $L$ are convex bodies, then:

$$h_{conv(K \bigcup L)}=max\{h_K, h_L\}$$

First doubt is: $h$ should be the support function, defined as: $h_C(u)=\sup\ \{ u \cdot y:y \in C \} \text{ for } u \in \mathbb{E}^d$, or not? And if yes, how do I apply it to a convex hull?

Answer 1

Fix $x$. Consider a subspace $ \mathbb{R}\cdot x$ When $f : \mathbb{E}^d\rightarrow \mathbb{R}\cdot x$ is orthogonal projection, then $h_K(x)$ is one of end points of interval $f(K)$

And from $f({\rm conv}\ K\cup L )= {\rm conv}\ f(K)\cup f(L) $ we can complete the proof.