Theorems · Theorem · geometry

EuclideanGeometry.exists_dist_eq_iff_exists_dist_orthogonalProjection_eq

∀ {𝕜 : Type u_1} {V : Type u_2} {P : Type u_3} [inst : RCLike 𝕜] [inst_1 : NormedAddCommGroup V]
  [inst_2 : InnerProductSpace 𝕜 V] [inst_3 : MetricSpace P] [inst_4 : NormedAddTorsor V P] {s : AffineSubspace 𝕜 P}
  [inst_5 : Nonempty ↥s] [inst_6 : s.direction.HasOrthogonalProjection] {ps : Set P},
  ps ⊆ ↑s →
    ∀ (p : P),
      (∃ r, ∀ p₁ ∈ ps, dist p₁ p = r) ↔ ∃ r, ∀ p₁ ∈ ps, dist p₁ ↑((EuclideanGeometry.orthogonalProjection s) p) = r

There exists r such that p has distance r from all the points of a set of points in s if and only if there exists (possibly different) r such that its orthogonalProjection has that distance from all the points in that set.

Defined in

Mathlib.Geometry.Euclidean.Projection

Cited by

3 results in Mathlib

Foundations

Depth 184 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RCLikeNormedAddCommGroupInnerProductSpaceMetricSpaceNormedAddTorsorNonemptySubmodule.HasOrthogonalProjection