# Can Minkowski difference be realized with support functions?

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.