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:
- 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.
- Gauge — I'm getting mixed up by the effect of $a$ in the denominator.
Best 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.