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Minkowski sum of two compact convex sets is easily computed if they are represented in terms of support functions, one just adds the two support vectors for each direction.

$X \oplus Y = \{x+y : x \in X \quad \mathrm{and}\quad y \in Y\}$

$\rho(l,X\oplus Y) = \rho(l,X) + \rho(l,Y)$ where $l \in \mathbb{R}^n$

Does this also hold for Minkowski difference of two compact convex sets?

$X \ominus Y = \{x-y : x \in X \quad \mathrm{and}\quad y \in Y\}$

Can one just take the difference of their respective support vectors? If not, are there conditions under which this will hold?

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$X \ominus Y = X \oplus (-Y)$. So yes.

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