The pointwise Clarkson inequality

For 2 ≤ q and vectors a, b of a normed group whose norm satisfies the parallelogram identity (InnerProductSpaceable), in particular of any real or complex inner product space,

‖a + b‖ ^ q + ‖a - b‖ ^ q ≤ 2 ^ (q - 1) * (‖a‖ ^ q + ‖b‖ ^ q).

At q = 2 this is the parallelogram identity, read as an inequality. For larger q it follows from the parallelogram identity, superadditivity of t ↦ t ^ (q / 2) on [0, ∞), and the power mean inequality (s + t) ^ r ≤ 2 ^ (r - 1) * (s ^ r + t ^ r). Integrating it gives Clarkson's inequality for Lᵖ spaces, 2 ≤ p < ∞, and with it their uniform convexity.

Main statements

• TauCeti.norm_add_rpow_add_norm_sub_rpow_le: the pointwise Clarkson inequality.

References

• J. A. Clarkson, Uniformly convex spaces, Trans. Amer. Math. Soc. 40 (1936), 396–414.
• H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Theorem 4.10.

theorem TauCeti.norm_add_rpow_add_norm_sub_rpow_le {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpaceable E] {q : ℝ} (hq : 2 ≤ q) (a b : E) : ‖a + b‖ ^ q + ‖a - b‖ ^ q ≤ 2 ^ (q - 1) * (‖a‖ ^ q + ‖b‖ ^ q)

The pointwise Clarkson inequality. For 2 ≤ q and vectors a, b of a normed group satisfying the parallelogram identity, ‖a + b‖ ^ q + ‖a - b‖ ^ q ≤ 2 ^ (q - 1) * (‖a‖ ^ q + ‖b‖ ^ q).