import Mathlib

namespace OAI

/-! Convex bodies and exact coverings by a single lattice in real coordinate space. -/

namespace SingleLatticeCovering

open MeasureTheory
open scoped Pointwise

abbrev Space (n : ℕ) := Fin n → ℝ

def IsConvexBody {n : ℕ} (K : Set (Space n)) : Prop :=
  IsCompact K ∧ Convex ℝ K ∧ (interior K).Nonempty

def LatticeCovers {n : ℕ} (K : Set (Space n)) (L : Submodule ℤ (Space n)) : Prop :=
  K + (L : Set (Space n)) = Set.univ

end SingleLatticeCovering

end OAI