Euclidean body
= Euclidean body
A Euclidean body is a nonempty bounded <Lebesgue measurable set> in $\mathbb R^n$ of positive <Lebesgue measure>. Projection inequalities require measurable coordinate projections, as for a <compact set> or a bounded <Borel set>. Modifying a body on a set of volume zero can change its coordinate-projection measures, so the actual projection sets matter. Geometric equality statements commonly impose additional regularity.