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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?

2 Answers 2

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

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It's not exactly the difference of the support function values, but you can take the difference of the support function value of l in X and -l in Y.

ρ(l,X⊖Y)=ρ(l,X) - ρ(-l,Y)

Here's a simple 1-d example showing the difference:

X = [0, 10]

Y = [0, 1]

X⊖Y = [-1, 10]

if l is the vector <1>, then ρ(l,X⊖Y) should be 10

However, ρ(l,X) = 10, ρ(l,Y) = 1, and so ρ(l,X) - ρ(l,Y) gives the incorrect answer 9.

Whereas ρ(l,X) - ρ(-l,Y) = 10 - 0 = 10, which is correct