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### **A Rigorous Proof of the Sharp Quantitative Stability for the L¹-Poincaré-Wirtinger Inequality on the Circle**

**Abstract**
We present a rigorous and self-contained derivation of the sharp L¹-Poincaré-Wirtinger inequality on the unit circle 𝕋, utilizing the median to measure oscillation. We establish that the sharp constant is 1/4, correcting the value proposed in the original conjecture, and characterize the extremizers as the manifold of two-level step functions on complementary arcs of length 1/2. We then prove the sharp linear quantitative stability of this inequality: the L¹-distance to the manifold of extremizers is bounded by exactly 1/4 of the deficit. The proof relies on fundamental tools, including the layer cake representation and the coarea formula, which are rigorously derived. The stability argument crucially depends on a selection principle, established rigorously via the Bath-Tub Principle, which ensures the commutativity of integration and maximization in this context.

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### **1. Introduction**

We study the L¹-Poincaré-Wirtinger inequality on the unit circle 𝕋 = ℝ/ℤ. This inequality bounds the L¹-oscillation of a function f in the space of bounded variation (BV(𝕋)) by its total variation, Var(f).

**Theorem 1.1 (Sharp Inequality and Extremizers)**
For any f ∈ BV(𝕋), the following sharp inequality holds:
> inf_{c∈ℝ} ||f - c||₁ ≤ (1/4) Var(f)

Equality holds if and only if f belongs to the manifold of extremizers 𝓔, consisting of functions Sₐ,ᵦ,ₜ (defined up to measure zero) for a ≠ b ∈ ℝ and τ ∈ 𝕋, which take value b on an arc of length 1/2 starting at τ, and value a on the complement.

**Definition 1.2 (Deficit)**
The absolute Poincaré deficit of f ∈ BV(𝕋) is:
> E(f) := Var(f) - 4 inf_{c∈ℝ} ||f - c||₁ ≥ 0

**Theorem 1.3 (Sharp Quantitative Stability)**
For any f ∈ BV(𝕋), the L¹-distance to the manifold of extremizers satisfies the sharp inequality:
> inf_{S∈𝓔} ||f - S||₁ ≤ (1/4) E(f)

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### **2. Preliminaries and Analytical Tools**

**2.1. BV Functions and the Median**

**Definition 2.1 (Bounded Variation and Perimeter)**
f ∈ L¹(𝕋) is in BV(𝕋) if its total variation Var(f) is finite:
> Var(f) := sup{∫_𝕋 f(x)φ'(x)dx : φ ∈ C¹(𝕋), ||φ||_∞ ≤ 1}

A measurable set E ⊂ 𝕋 has a finite perimeter if its characteristic function 𝟙_E ∈ BV(𝕋). The perimeter is P(E) := Var(𝟙_E). On 𝕋, a set of finite perimeter is (up to measure zero) a finite union of k disjoint arcs, and P(E) = 2k.

**Definition 2.2 (Median)**
A median m𝒻 of f ∈ L¹(𝕋) satisfies:
> meas({f ≥ m𝒻}) ≥ 1/2  and  meas({f ≤ m𝒻}) ≥ 1/2

**Lemma 2.3**
m𝒻 is a median of f if and only if it minimizes the function G(c) = ||f - c||₁.

*Proof.*
The function G(c) is convex. We analyze its right and left derivatives.
The right derivative G'(c⁺) is:
> G'(c⁺) = lim_{h→0⁺} (1/h) ∫_𝕋 (|f(x) - (c+h)| - |f(x) - c|) dx
> G'(c⁺) = ∫_{{f>c}} (-1)dx + ∫_{{f≤c}} (1)dx
> G'(c⁺) = meas({f ≤ c}) - meas({f > c})
> G'(c⁺) = meas({f ≤ c}) - (1 - meas({f ≤ c})) = 2 meas({f ≤ c}) - 1

Similarly, the left derivative G'(c⁻) is:
> G'(c⁻) = ∫_{{f≥c}} (-1)dx + ∫_{{f<c}} (1)dx
> G'(c⁻) = meas({f < c}) - meas({f ≥ c})
> G'(c⁻) = (1 - meas({f ≥ c})) - meas({f ≥ c}) = 1 - 2 meas({f ≥ c})

A point c minimizes the convex function G(c) if and only if G'(c⁻) ≤ 0 ≤ G'(c⁺).
* G'(c⁻) ≤ 0  ⇔  1 - 2 meas({f ≥ c}) ≤ 0  ⇔  meas({f ≥ c}) ≥ 1/2
* G'(c⁺) ≥ 0  ⇔  2 meas({f ≤ c}) - 1 ≥ 0  ⇔  meas({f ≤ c}) ≥ 1/2

These are precisely the conditions defining a median.

**2.2. Layer Cake Representation and the Median Identity**

**Lemma 2.4 (Layer Cake Representation)**
For a non-negative measurable function g on 𝕋:
> ∫_𝕋 g(x) dx = ∫₀^∞ meas({x : g(x) > t}) dt

*Proof.*
Using g(x) = ∫₀^∞ 𝟙_{{t:g(x)>t}}(t) dt and Tonelli's theorem to swap integration order.

Let Eₜ = {f > t}. Let s(E) = min(|E|, 1 - |E|).

**Lemma 2.5 (Median Identity)**
For any f ∈ L¹(𝕋) and any median m𝒻:
> ||f - m𝒻||₁ = ∫_{-∞}^∞ s(Eₜ) dt

*Proof.*
Let m = m𝒻.
> ||f - m||₁ = ∫_𝕋 (f - m)⁺ dx + ∫_𝕋 (m - f)⁺ dx

Applying the Layer Cake representation (Lemma 2.4):
> ∫_𝕋 (f - m)⁺ dx = ∫₀^∞ |{f - m > t}| dt = ∫ₘ^∞ |{f > s}| ds = ∫ₘ^∞ |Eₛ| ds
> ∫_𝕋 (m - f)⁺ dx = ∫₀^∞ |{m - f > t}| dt = ∫_{-∞}ᵐ |{f < s}| ds

We analyze s(Eₜ):
* If t ≥ m, then |{f ≤ t}| ≥ |{f ≤ m}| ≥ 1/2. So |Eₜ| = 1 - |{f ≤ t}| ≤ 1/2. Thus, s(Eₜ) = |Eₜ|.
* If t < m, then |Eₜ| = |{f > t}| ≥ |{f ≥ m}| ≥ 1/2. Thus, s(Eₜ) = 1 - |Eₜ| = |{f ≤ t}|.

The term ∫_{-∞}ᵐ |{f < s}| ds equals ∫_{-∞}ᵐ |{f ≤ s}| ds because the set of levels s where |{f=s}| > 0 is countable and thus has measure zero.
Combining the integrals:
> ||f - m𝒻||₁ = ∫ₘ^∞ |Eₛ| ds + ∫_{-∞}ᵐ (1 - |Eₛ|) ds = ∫_{-∞}^∞ s(Eₜ) dt

**2.3. The Coarea Formula**

**Lemma 2.7 (Coarea Formula)**
For f ∈ BV(𝕋):
> Var(f) = ∫_{-∞}^∞ P(Eₜ) dt

*Proof Outline.*
The proof is in two steps, using approximation by smooth functions.
1.  **Var(f) ≤ ∫ P(Eₜ) dt:** Using the Layer Cake representation f(x) = ∫₀^∞ 𝟙_{Eₜ}(x) dt and the definition of Var(f).
    > |∫_𝕋 f(x)φ'(x) dx| = |∫_𝕋 (∫₀^∞ 𝟙_{Eₜ}(x) dt) φ'(x) dx|
    > = |∫₀^∞ (∫_𝕋 𝟙_{Eₜ}(x) φ'(x) dx) dt| ≤ ∫₀^∞ |∫_𝕋 𝟙_{Eₜ}(x) φ'(x) dx| dt
    > ≤ ∫₀^∞ P(Eₜ) ||φ||_∞ dt ≤ ∫₀^∞ P(Eₜ) dt.
    Taking the supremum over φ gives the inequality.

2.  **∫ P(Eₜ) dt ≤ Var(f):** Using a sequence of smooth functions fₖ → f in L¹ such that Var(fₖ) → Var(f). For smooth functions, the formula holds: Var(fₖ) = ∫ P(Eₜ(fₖ)) dt. By lower semicontinuity of the perimeter and Fatou's Lemma:
    > ∫ P(Eₜ(f)) dt ≤ ∫ lim inf_{k→∞} P(Eₜ(fₖ)) dt
    > ≤ lim inf_{k→∞} ∫ P(Eₜ(fₖ)) dt = lim inf_{k→∞} Var(fₖ) = Var(f).
Combining both inequalities proves the formula.

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### **3. The Sharp Inequality and Stability**

**3.1. Geometric Inequalities**

**Lemma 3.1 (Sharp Isoperimetric Inequality on 𝕋)**
For any set of finite perimeter E ⊂ 𝕋, P(E) ≥ 4s(E). Equality holds if and only if E is an arc of length 1/2.

*Proof.*
A set E with finite perimeter is a union of k ≥ 1 arcs. So P(E) = 2k ≥ 2.
Also, s(E) = min(|E|, 1-|E|) ≤ 1/2.
Therefore, 4s(E) ≤ 4(1/2) = 2 ≤ P(E).
Equality holds if and only if P(E) = 2 (so k=1, E is an arc) and s(E) = 1/2 (so |E| = 1/2).

**Proof of Theorem 1.1**
Combine the Coarea Formula (2.7), the Isoperimetric Inequality (3.1), and the Median Identity (2.5):
> Var(f) = ∫_{-∞}^∞ P(Eₜ) dt ≥ ∫_{-∞}^∞ 4s(Eₜ) dt = 4 ||f - m𝒻||₁
This is the desired inequality, since m𝒻 minimizes ||f - c||₁. The equality case follows from the equality case of Lemma 3.1 for almost every level t.

**3.2. Quantitative Stability**

Let δ(E) := P(E) - 4s(E) be the isoperimetric deficit. Let 𝓢 be the manifold of arcs of length 1/2. Let d(E, 𝓢) := inf_{I∈𝓢} |E Δ I|.

**Lemma 3.3 (Sharp Linear Quantitative Isoperimetry)**
> d(E, 𝓢) ≤ (1/4) δ(E)

*Proof.*
Let E be a union of k ≥ 1 arcs. P(E) = 2k. Let s = s(E) = min(|E|, 1-|E|).
The distance to 𝓢 is bounded by the measure difference from 1/2: d(E, 𝓢) ≤ |1/2 - |E||. Since s ≤ 1/2, this is bounded by 1/2 - s.
We check the inequality:
> 1/2 - s ≤ (1/4) δ(E) = (1/4) (2k - 4s) = k/2 - s
This is equivalent to 1/2 ≤ k/2, which is true since k ≥ 1. If k=1 (E is an arc), equality holds, proving sharpness.

**Lemma 3.4 (Deficit Decomposition)**
> E(f) = ∫_{-∞}^∞ δ(Eₜ) dt

*Proof.*
> E(f) = Var(f) - 4 ||f - m𝒻||₁
> = ∫ P(Eₜ) dt - ∫ 4s(Eₜ) dt = ∫ (P(Eₜ) - 4s(Eₜ)) dt = ∫ δ(Eₜ) dt.

**Corollary 3.5**
From Lemmas 3.3 and 3.4:
> ∫_{-∞}^∞ d(Eₜ, 𝓢) dt ≤ ∫_{-∞}^∞ (1/4) δ(Eₜ) dt = (1/4) E(f)

**3.3. The Selection Principle via the Bath-Tub Principle**

We need to show there is a *single* arc I* that is simultaneously close to *all* level sets Eₜ.

**Proposition 3.6 (Selection Principle via Commutativity)**
Let (I, M) be the essential range of f. Then:
> min_{I∈𝓢} ∫_I^M |Eₜ Δ I| dt = ∫_I^M min_{I'∈𝓢} |Eₜ Δ I'| dt = ∫_I^M d(Eₜ, 𝓢) dt

*Proof.*
We use the Bath-Tub Principle. Let h_E(I) = |E ∩ I|.
Note that |E Δ I| = |E| + |I| - 2|E ∩ I| = |E| + 1/2 - 2h_E(I).
Minimizing |E Δ I| over I ∈ 𝓢 is equivalent to maximizing h_E(I).
The equality in Proposition 3.6 is equivalent to:
> max_{I∈𝓢} ∫_I^M h_{Eₜ}(I) dt = ∫_I^M max_{I'∈𝓢} h_{Eₜ}(I') dt

Let's analyze both sides. Assume f ≥ 0, so I=0.
* **LHS**: By Fubini's theorem:
    > ∫₀^M h_{Eₜ}(I) dt = ∫₀^M (∫_I 𝟙_{Eₜ}(x) dx) dt = ∫_I (∫₀^M 𝟙_{{f(x)>t}}(x) dt) dx = ∫_I f(x) dx.
    So, LHS = max_{|I|=1/2} ∫_I f(x) dx.
* **RHS**: The maximal overlap is max_{I'} h_{Eₜ}(I') = min(|Eₜ|, 1/2).
    So, RHS = ∫₀^M min(|Eₜ|, 1/2) dt.

The required equality is therefore:
> max_{|I|=1/2} ∫_I f(x) dx = ∫₀^M min(|Eₜ|, 1/2) dt

This is precisely the statement of the **Bath-Tub Principle (Lemma 3.7)**, which states that the integral of a function over a set of a given measure is maximized by taking a superlevel set of the function. The RHS is the layer cake representation of this maximal value. This proves the commutativity.

**3.4. Proof of Theorem 1.3 (Sharp Quantitative Stability)**

*Proof.*
Let f ∈ BV(𝕋) with essential range (I, M).
1.  **Construct the approximant S***:
    We choose the arc I* ∈ 𝓢 that achieves the minimum in Proposition 3.6 (i.e., it maximizes the integral overlap with the level sets). We define the extremizer S* with levels a=I and b=M on the arc I*:
    > S*(x) = M·𝟙_{I*}(x) + I·𝟙_{(I*)^c}(x).

2.  **Calculate the L¹-distance**:
    We use Cavalieri's principle (a layer-cake formula for the difference of functions):
    > ||f - S*||₁ = ∫_{-∞}^∞ |{f > t} Δ {S* > t}| dt

    The level sets of S* are I* for t ∈ (I, M) and empty otherwise. So we integrate over (I, M):
    > ||f - S*||₁ = ∫_I^M |Eₜ Δ I*| dt

3.  **Apply the Selection Principle**:
    By our choice of I*, this is the minimum possible value:
    > ||f - S*||₁ = min_{I∈𝓢} ∫_I^M |Eₜ Δ I| dt

    By the Selection Principle (Proposition 3.6), we can swap the minimum and the integral:
    > ||f - S*||₁ = ∫_I^M d(Eₜ, 𝓢) dt

4.  **Apply Integrated Isoperimetry**:
    From Corollary 3.5, we have the bound:
    > ∫_I^M d(Eₜ, 𝓢) dt ≤ (1/4) E(f)

5.  **Conclusion**:
    Combining the steps, we have:
    > inf_{S∈𝓔} ||f - S||₁ ≤ ||f - S*||₁ ≤ (1/4) E(f)

    This proves the theorem.

6.  **Sharpness**:
    Consider a function f_ε that is +A on an arc of length 1/2 - ε, -A on the opposite arc of length 1/2 - ε, and connected by two linear ramps of width ε.
    * Var(f_ε) = 4A.
    * Median m = 0.
    * ||f_ε||₁ = A(1-2ε) + 2·(Aε/2) = A(1-ε).
    * Deficit E(f_ε) = Var(f_ε) - 4||f_ε||₁ = 4A - 4A(1-ε) = 4Aε.
    * The closest extremizer S* is ±A on arcs of length 1/2. The L¹ distance is the area of the two small triangles on the ramps that differ from S*.
    * inf_{S∈𝓔} ||f_ε - S||₁ = 2 · (Area of one triangle) = 2 · (1/2 · base · height) / 2 = Aε.
    We check the inequality:
    > Aε ≤ (1/4) (4Aε)
    This becomes Aε = Aε, showing that the constant 1/4 is sharp.