Minkowski functional and atomic norm

Question

We define the convex hull of an algebraic set $A$. When a basic semi-algebraic set $A \subset R_p$ is bounded, we denote its convex hull as follows: $C(A) = \mathrm{conv}(A)$. We defined the Minkowski-functional by $p(x):= \inf\{ \lambda >0 \mid x \in \lambda C(A) \}$. My question is why: $$ \inf\{ \lambda >0:x \in \lambda C(A)\}= \inf\left\{ \sum_{\bf{a}\in A}c_a: x = \sum_{\bf{a} \in A} c_a\bf{a}, \; c_a \ge 0, \ \forall\bf{a}\in A\right\}. $$

Answer 1

Recall that a point $z$ lies in the convex hull of $A$, $z\in C(A)$, iff we can write,

$$z = \sum\limits_{a\in A}z_a a$$

with $\sum\limits_{z\in A}z_A = 1$ and $z_a\geq 0$ for all $a\in A$.

Then the set $\lambda C(A)$ is the set of all points $\lambda z$, $z\in C(A)$. Since $z\in C(A)$, we can write,

$$\lambda z = \lambda \sum\limits_{a\in A} z_a a$$

and since $\sum\limits_{a\in A}z_a=1\implies \lambda\sum\limits_{a\in A}z_a=\lambda$ we are done.