Ok, I've got this exercise:
Let $K$ be a convex body in $\mathbb{R}^n$ and let $\phi: K \to \mathbb{R}$ be a cone function. Set $M := \text{max}_{x\in K}\phi (x)$. Prove that, for $0<t<M$, it holds $\{\phi \geq t \}=(1-(\frac{t}{M}))K+a$, where $a$ is some vector that may depend on $t$.
Now, after a suggestion, I translate $K$ and $\phi$ appropriately so that $\phi (0) = \text{max}_{x\in K}\phi(x)$, so $a$ is always $0$.
What's next?