Let $K \subset R^d $ be an compact convex set with $0 \in int (K)$ let: $$h_K(x) = \max_{y \in K } \langle x,y\rangle$$ be the support function of $K$
the Minkowski functional of $K$ is define as
$$\|x\|_K = \min \{\lambda \geq 1: x \in \lambda K\}$$
and
$$K^* = \{y \in R^d : \forall x \in K, \langle x,y\rangle \leq 1 \}$$ be the polar set of $K$ then $h_K(\cdot) = \|\cdot\|_{K^*}$
how to see that?
As noted in the comments, the Minkowski functional is written with the infimum over $\lambda >0$, or else this is false. For any $\alpha>0$ and $x\in \mathbb{R}^d$, \begin{gather*} \alpha\geq h_K(x)\iff\\ \alpha\geq \langle x,y\rangle\quad \forall y\in K\iff\\1\geq \langle \frac{1}{\alpha}x,y\rangle \quad \forall y\in K\iff\\ \frac{1}{\alpha}x\in K^*\iff\\ x\in \alpha K^*\iff\\ \alpha\geq \|x\|_{K^*}. \end{gather*} It is not hard to see that $h_K(x)$ and $\|x\|_K^*$ are finite and nonnegative by the compactness and $0$ in the interior assumptions, and this shows they have the same set of upper bounds, and therefore are equal.