# Explaining Defitions of Minkowski functional and Gauge Functional

I'm having trouble understanding the definition of Minkowski functional.

Let $$K$$ be a symmetric (i.e. if it contains $$x$$ it also contains $$-x$$) convex body in a linear space $$V$$. We define a function $$p$$ on $$V$$ as $$p(x) = \inf \{ \lambda \in \mathbb{R}_{> 0} : x \in \lambda K \}$$ Where $$p$$ is called the Minkowski functional.

Lax's gives the defintion of gauge (with respect to origin) as follows:

If $$K \subset V$$ is a convex set in a vector space with an interior point, the gauge $$p_K$$ is given by: $$p_K(x) = \inf a \quad a>0,\frac{x}{a} \in K$$

I'm having trouble breaking down the definitions in "plain english". My attempt:

1. Minkowski functional -- choose a point in $$K$$. Take all reals, $$\lambda$$. $$\lambda K$$ is then a "scaled version of $$K$$". $$p(x)$$ is the "smallest" $$\lambda$$ such that $$x$$ is still in $$\lambda K$$. If I think about a unit ball at the origin and $$x$$ being the origin, is $$p(x) = 0$$? The set of all $$\lambda$$ , according to my understanding, will be all positive reals - the inf of which is zero.
2. Gauge -- I'm getting mixed up by the effect of $$a$$ in the denominator.

## 1 Answer

First of all, I think the key to the trouble of your understanding is the following relation: $$\forall a>0\; \frac{x}{a} \in K \iff x \in aK$$ So the definition you wrote for the Minkowski functional and the gauge Lax defines are essentially the same, note that you missed some key esssential properties: In the definition for the Minkowski functional you have the assume that $$0 \in K$$ is an interior point (we call such sets absorbing), since only then every vector $$v \in V$$ has some scalar small enough such that $$v \in \lambda K$$, otherwise the infimum maybe empty. The same thing goes for gauge: Lax explicitly mentions that he assumes the interior point is $$0$$.

Note that your intuition in 1) is reasoable (yes $$p(0)=0$$), but we are not only talking about points in $$K$$, but in the whole of $$V$$, this is used in the Geometric Version of the Hahn Banach Theorem for example.