convex_superlevel · Samplinglib

Plain-English statement

- Superlevel sets of a positive log-concave function are convex within the log-concavity domain.

Scope guard. This card records a compiled local declaration. Its mathematical scope is exactly the Lean statement below; the Registry note and source correspondence may describe motivation but do not strengthen it.

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Proof architecture

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Lean statement

theorem LogConcaveOn.convex_superlevel {E : Type*} [AddCommMonoid E] [Module ℝ E]
    {s : Set E} {f : E → ℝ}
    (hf : LogConcaveOn s f) (c : ℝ) :
    Convex ℝ {x ∈ s | c ≤ f x} := by
  by_cases hc : c ≤ 0
  · intro x hx y hy a b ha hb hab
    have hmid_s := hf.convex_domain hx.1 hy.1 ha hb hab
    exact ⟨hmid_s, le_of_lt (lt_of_le_of_lt hc (hf.pos (x := a • x + b • y) hmid_s))⟩
  · have hcpos : 0 < c := lt_of_not_ge hc
    intro x hx y hy a b ha hb hab
    have hmid_s := hf.convex_domain hx.1 hy.1 ha hb hab
    refine ⟨hmid_s, ?_⟩
    have hlogx : Real.log c ≤ Real.log (f x) :=
      Real.log_le_log hcpos hx.2
    have hlogy : Real.log c ≤ Real.log (f y) :=
      Real.log_le_log hcpos hy.2
    have hweighted :
        a * Real.log c + b * Real.log c ≤
          a * Real.log (f x) + b * Real.log (f y) :=
      add_le_add (mul_le_mul_of_nonneg_left hlogx ha) (mul_le_mul_of_nonneg_left hlogy hb)
    have hconst : a * Real.log c + b * Real.log c = Real.log c := by
      calc
        a * Real.log c + b * Real.log c = (a + b) * Real.log c := by ring
        _ = Real.log c := by rw [hab]; ring
    have hfloor : Real.log c ≤ a * Real.log (f x) + b * Real.log (f y) := by
      simpa [hconst] using hweighted
    have hconc :
        a * Real.log (f x) + b * Real.log (f y) ≤
          Real.log (f (a • x + b • y)) := by
      simpa [smul_eq_mul] using hf.concaveOn_log.2 hx.1 hy.1 ha hb hab
    exact (Real.log_le_log_iff hcpos (hf.pos (x := a • x + b • y) hmid_s)).mp
      (hfloor.trans hconc)

/-- Positive log-concave functions are quasiconcave: all superlevel sets are convex. -/

Open AutoSamplingTheory/TechnicalLemmas/Geometry/LogConcavity.lean:82published source at 7bcd37294df1

Proof architecture

Chewi CONV/DENS root: identify positive-density superlevel sets as convex bodies for restrictions and localization

Lean proof walkthrough

Read the quantified variables and typeclass brackets as part of the mathematical statement; inferred arguments are not missing assumptions.
`intro` introduces quantified hypotheses into the local proof context.
`have` creates a named intermediate mathematical fact.
`calc` records an equality or inequality chain matching a paper calculation.
`rw` rewrites by an established identity.
`refine` instantiates a reusable theorem while leaving explicit subgoals.
`exact` closes the current goal with an already typed term.
`simpa` closes the goal after a controlled simplification of a typed result.

Why the statement has this shape

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Hidden assumptions and non-claims

Density statements retain normalization and absolute-continuity prerequisites.