Plain-English statement
A positive real-valued function is log-concave on a set when its logarithm is concave there.
Mathematical statement
LogConcaveOn(s,f) means (forall x in s, 0 < f(x)) and log(f) is concave on s.
Intuition
Taking logarithms turns multiplicative interpolation inequalities for densities into ordinary additive concavity. ASTIS keeps positivity in the definition because the real logarithm alone would otherwise hide the density domain.
Conditions
- The domain is a real module with additive structure.
- Strict positivity holds at every point of the stated set.
- Concavity is relative to the same set and scalar field.
Why these conditions cannot be dropped
- Without positivity, Real.log is totalized in Lean and no longer expresses the intended density logarithm.
- Without a convex domain, Mathlib's ConcaveOn package cannot supply the interpolation consequences used later.
Proof route
- This is the canonical ASTIS definition.
- Projection lemmas expose positivity and concavity separately.
- Closure lemmas derive superlevel convexity, products, powers, and affine pullbacks.
Lean interface notes
- The definition is a conjunction, so .1 and .2 are the primitive interfaces.
- Set membership is explicit in positivity and ConcaveOn.
- Real.log is total; the strict positivity conjunct is mathematically essential.
Read the mathematics first, then descend into Lean
You do not need to know Lean before opening this panel. The page keeps the paper-level theorem, rigorous proof obligations, exact Lean declaration, and proof dependencies as separate layers so a first-time reader can move down one layer at a time.
Proof architecture
Start with the tree when learning: prerequisites sit below the theorem and downstream results sit above it. Switch to the network when you want to understand where this declaration lives in the local formal library. Every mapped node is clickable.
Graph rule: only dependencies found by the ASTIS source scan are drawn. Missing tactic indirection is treated as an under-approximation; the site never invents an edge just to make a prettier graph.
How to read the exact Lean declaration
Read a Lean theorem left-to-right exactly as you would unpack a mathematical sentence: name → ambient types → automatically inferred structures → explicit hypotheses → conclusion → proof. Then read the proof top-to-bottom as transformations of the current goal.
- NameWhat reusable mathematical fact is being created?
- ParametersWhich symbols are arbitrary, and which structures are inferred by typeclass search?
- PropositionAfter the colon, translate the Lean expression back into a paper statement.
- Proof actionsAfter
by, ask what each tactic does to the mathematical goal—not only what syntax it uses.
Syntax used on this page
This glossary is filtered to syntax that actually occurs in the declaration above. Open a symbol only when you meet it, rather than memorizing Lean grammar in advance.
Source voice and ASTIS voice stay separate
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Lean statement
def LogConcaveOn {E : Type*} [AddCommMonoid E] [Module ℝ E]
(s : Set E) (f : E → ℝ) : Prop :=
(∀ x ∈ s, 0 < f x) ∧ ConcaveOn ℝ s (fun x => Real.log (f x))Open AutoSamplingTheory/TechnicalLemmas/Geometry/LogConcavity.lean:29published source at 7bcd37294df1