Manuscript Oai276-A@1 · Oai276-A.tex · lemma · 215068af

Lemma: Scalar convexity

Local one-use proof › Elementary entropy facts · lines 411–461 · proof included or linked

Oai276-A@1#lem:convex

The function $s \mapsto g_{2} (s^{2})$ is nondecreasing and convex on $[0, 1/2]$ . Consequently,

e_{γ, ν} (p) := g_{2} (v_{γ, ν} (p))

is convex on $[0, 1]$ for every $γ, ν \in [0, 1]$ .

Proof. It suffices to calculate in nats. For $0 < u < 1/4$ , set

r = 1 - 4 u, L = \frac{atanh r}{r} .

Differentiation gives

g_{e}^{'} (u) = 2 L, u g_{e}^{''} (u) + \frac{1}{2} g_{e}^{'} (u) = \frac{L - 1}{r ^{2}} \geq 0.

The inequality follows from $atanh r \geq r$ . Therefore

\frac{d}{d s} g_{e} (s^{2}) \geq 0, \frac{d ^{2}}{d s ^{2}} g_{e} (s^{2}) = 4 (s^{2} g_{e}^{''} (s^{2}) + \frac{1}{2} g_{e}^{'} (s^{2})) \geq 0.

Continuous endpoint limits extend the conclusion to the closed interval.

Finally,

v_{γ, ν} (p) = (γ ν (1 - ν) γ (1 - γ) (p - ν))_{2}

is convex. Compose it with the nondecreasing convex function just proved.

Displayed formulas in this passage

unlabeled · line 416
$e_{γ, ν} (p) := g_{2} (v_{γ, ν} (p))$
unlabeled · line 425
$r = 1 - 4 u, L = \frac{atanh r}{r} .$
unlabeled · line 431
$g_{e}^{'} (u) = 2 L, u g_{e}^{''} (u) + \frac{1}{2} g_{e}^{'} (u) = \frac{L - 1}{r ^{2}} \geq 0.$
unlabeled · line 438
$\frac{d}{d s} g_{e} (s^{2}) \geq 0, \frac{d ^{2}}{d s ^{2}} g_{e} (s^{2}) = 4 (s^{2} g_{e}^{''} (s^{2}) + \frac{1}{2} g_{e}^{'} (s^{2})) \geq 0.$
unlabeled · line 449
$v_{γ, ν} (p) = (γ ν (1 - ν) γ (1 - γ) (p - ν))_{2}$