File size: 1,316 Bytes
cead56f | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 | import Mathlib.Analysis.InnerProductSpace.Basic
namespace OAI
noncomputable section
open Set
open scoped RealInnerProductSpace
namespace ProjectionCounterexample
/-- For a positively homogeneous support bound, unit directions determine all
directions. This connects the unit-sphere definition with an
explicit zonotope's halfspace description. -/
theorem unit_support_halfspaces_eq
{V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℝ V]
(f : V → ℝ) (hzero : 0 ≤ f 0)
(hscale : ∀ (a : ℝ), 0 ≤ a → ∀ u, f (a • u) = a * f u) :
{x : V | ∀ u, ‖u‖ = 1 → ⟪u, x⟫ ≤ f u} =
{x : V | ∀ u, ⟪u, x⟫ ≤ f u} := by
ext x
constructor
· intro hx u
by_cases hu : u = 0
· simpa [hu] using hzero
have hnorm : ‖u‖ ≠ 0 := norm_ne_zero_iff.mpr hu
let v : V := ‖u‖⁻¹ • u
have hv : ‖v‖ = 1 := by
simp [v, norm_smul, hnorm]
have huv : ‖u‖ • v = u := by
simp [v, smul_smul, hnorm]
calc
⟪u, x⟫ = ‖u‖ * ⟪v, x⟫ := by
rw [← real_inner_smul_left, huv]
_ ≤ ‖u‖ * f v := mul_le_mul_of_nonneg_left (hx v hv) (norm_nonneg u)
_ = f u := by rw [← hscale ‖u‖ (norm_nonneg u) v, huv]
· intro hx u _
exact hx u
end ProjectionCounterexample
end
end OAI
|