For a compact convex set \( K \subset \mathbb{R}^2 \), the support function \( h_K \) is defined for a direction \( \theta \) as:
\[
h_K(\theta) = \max_{x \in K} \langle x, \theta \rangle
\]
where \( \langle x, \theta \rangle \) denotes the dot product of the
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