From: Jakub Kądziołka <kuba@kadziolka.net>

Hello,

I have stumbled upon this lemma in HOL-Analysis.Convex:

lemma convex_on_alt:

fixes C :: "'a::real_vector set"

assumes "convex C"

shows "convex_on C f ⟷

(∀x ∈ C. ∀ y ∈ C. ∀ μ :: real. μ ≥ 0 ∧ μ ≤ 1 ⟶

f (μ *⇩R x + (1 - μ) *⇩R y) ≤ μ * f x + (1 - μ) * f y)"

As this is a simple restatement of convex_on_def, I was somewhat puzzled

by the assumption of "convex C". As it turns out - it's not necessary.

I removed the assumes line and the proof succeeded without any

modification. I suppose it would be useful to reflect this in the

distribution?

Kind regards,

Jakub Kądziołka

Last updated: Jul 15 2022 at 23:21 UTC