Editing Minkowski addition (section)

== Essential Minkowski sum ==
There is also a notion of the '''essential Minkowski sum''' +<sub>e</sub> of two subsets of Euclidean space. The usual Minkowski sum can be written as

<math display="block">A + B = \left\{ z \in \mathbb{R}^{n} \,|\, A \cap (z - B) \neq \emptyset \right\}.</math>

Thus, the '''essential Minkowski sum''' is defined by

<math display="block">A +_{\mathrm{e}} B = \left\{ z \in \mathbb{R}^{n} \,|\, \mu \left[A \cap (z - B)\right] > 0 \right\},</math>

where ''μ'' denotes the ''n''-dimensional [[Lebesgue measure]]. The reason for the term "essential" is the following property of [[indicator function]]s: while

<math display="block">1_{A \,+\, B} (z) = \sup_{x \,\in\, \mathbb{R}^{n}} 1_{A} (x) 1_{B} (z - x),</math>

it can be seen that

<math display="block">1_{A \,+_{\mathrm{e}}\, B} (z) = \mathop{\mathrm{ess\,sup}}_{x \,\in\, \mathbb{R}^{n}} 1_{A} (x) 1_{B} (z - x),</math>

where "ess&thinsp;sup" denotes the [[essential supremum]].