Optimality of Gerver's Sofa

1.1 Introduction

• Gerver’s sofa G attains the maximum area αmax of a moving sofa.The proof requires no computer assistance beyond numerical calculations possible with a scientific calculator.Source: §1.1 · p. 6

1.1. INTRODUCTION

• The argument first restricts a maximum-area sofa to be monotone, balanced, and rotating through the full right angle, then proves its injectivity condition.Injectivity states that the hallway’s inner-corner trajectory has no self-loops in the sofa’s frame and follows from a new differential inequality inspired by Romik’s ODE [Rom18].Source: §1.1 · p. 7
• The proof reduces sofa-area maximization to a concave quadratic functional Q whose global maximum is attained by G.Injectivity makes the enclosing boundary a Jordan curve, enabling Green’s theorem; Brunn–Minkowski theory and Mamikon’s theorem establish the upper bound and concavity, while Romik’s local optimality equations [Rom18] certify G.Source: §1.1 · pp. 7–8

1.2. MONOTONE SOFAS AND CAPS

• The monotone-sofa reduction addresses a connectedness gap in earlier intersection-based derivations [KR18].Earlier works did not rigorously establish that the full intersection was connected or equal to a maximum-area sofa, whereas the supporting-hallway construction proves these properties.Source: §1.2 · p. 11, §1.2.2 · p. 11

1.2 Monotone Sofas and Caps

• A maximum-area sofa can be assumed monotone, meaning it equals the connected intersection I(S) of its continuously chosen supporting hallways.The supporting-hallway construction proves I(S) remains connected, moves continuously inside the hallway, contains S, and satisfies I(S)=S when S is already monotone.Source: §1.2 · p. 11

1.2.2 Cap and Niche

• For a monotone sofa S, the cap K is a convex body inscribed in the parallelogram Pω, and S is obtained by subtracting the niche N(K) from K.The cap is determined by the supporting hallways’ outer walls, while the niche is the union of triangular regions carved out by their inner walls.Source: §1.2.2 · pp. 11–12
• A monotone sofa is represented as S = K \ N(K), so its area becomes the cap functional Aω(K) = |K| − |N(K)|.The niche is determined by the cap, and the paper proves niche containment to recover connectedness.Source: §1.2.2 · p. 12

1.3.1 Balancing Argument

• Gerver’s theorem [Ger92] asserts that a maximum-area moving sofa can be approximated arbitrarily closely by balanced polygons formed from increasingly dense rotated hallway intersections.The balancing argument translates a contributing strip or hallway to increase area whenever opposite parallel side lengths differ.Source: §1.3 · pp. 13–14

1.3.2 Logical Gap

• Gerver’s balancing argument has a logical gap: it does not establish connectedness of the polygon approximants or their limit, and balancing can disconnect an intersection.For Θ = {π/6, π/3} and ω = π/2, continuing the prescribed translation past a boundary makes SΘ disconnected.Source: §1.3.2 · p. 14

1.4.1 Limit of Maximum Polygon Caps

• The paper repairs Gerver’s proof by maximizing a cap functional over possibly disconnected polygon sofas, then proving niche containment before taking limits.This avoids assuming connectedness during the balancing step while ultimately obtaining a connected limiting sofa.Source: §1.4.1 · p. 16
• Maximum polygon caps converge to a balanced maximum sofa that is connected, monotone, and maximum-area among monotone sofas with rotation angle ω.The construction establishes N(Kω) ⊆ Kω and preserves maximization of Aω(Kω) = |Kω| − |N(Kω)| in the limit.Source: §1.4 · p. 15, §1.4.1 · p. 16

1.4.2 Balancedness of Maximum Polygon Cap

• Maximum polygon caps balance the side lengths of the cap and niche in opposite normal directions, with niche containment obtained as a consequence.The proof uses support functions and extends the polygon area functional to a larger function space.Source: §1.4.2 · p. 16

1.5.1 Statement

• Theorem 1.5.1 bounds the rotation angle of any moving sofa of area at least 2.2 by sec−1(2.2) ≤ω≤π/2.The lower bound follows from intersections with rotated hallways, while movements with ω > π/2 can be replaced by one reaching π/2.Source: §1.5.1 · p. 18
• A balanced maximum sofa with area at least 2.2 admits a movement with rotation angle ω = π/2.Theorem 1.5.2 extends the movement by first rotating a copy through π/2 −ω inside the horizontal hallway, then following its original movement.Source: §1.5.1 · p. 18

1.5.2 Proof Outline

• The key geometric step proves that the sofa Sω is disjoint from the triangular region ∆ω, leaving room for rotation by π/2 −ω inside the horizontal hallway.Balancedness equalizes the horizontal side lengths; two suitably separated boundary points and a supporting hallway then enclose ∆ω outside the sofa.Source: §1.5.1 · p. 18, §1.5.2 · p. 19

1.6 Surface Area Measure

• The surface area measure σK records the lengths of a planar convex body’s edges by their outward normal angles and acts as a weak derivative of its possibly non-differentiable boundary.It is defined on S1, measures polygonal edge lengths at singleton angles, and represents differential side lengths for smooth boundaries.Source: §1.6 · p. 20

1.7.1 Statement

• The injectivity condition states that the rotation path x of any balanced maximum sofa with rotation angle π/2 is continuously differentiable, with x′(t) · ut < 0 and x′(t) · vt > 0.These inequalities prevent self-intersection and make the trajectory a Jordan arc whose enclosed area can be expressed using Green’s theorem.Source: §1.7 · p. 21, §1.7.1 · pp. 21–22

1.7.2 A Differential Inequality

• The differential inequality generalizes Romik’s balancing ODE by bounding differential side lengths even when the supporting hallway lacks contact at B(t) or x(t).The proof approximates the sofa by balanced polygonal sofas, estimates sides on the right inner wall, and passes to the limit as the angle spacing tends to zero.Source: §1.7.2 · p. 22, §1.7 · p. 23

1.7. INJECTIVITY CONDITION

• The proof uses a reformulated inequality that remains valid when the cap has nondifferentiable or multiple contact points with the outer wall.The reformulation is derived from the earlier inequality and accounts for the general case through k0.Source: §1.7 · pp. 23–24

1.7.3 Solving the Differential Inequality

• The injectivity proof rewrites Equation (1.10) in terms of arm lengths f(t) and g(t), whose lower bounds can be iteratively improved.The function m0(x) = x − k0(x) is monotone increasing, supporting the lower-bound iteration.Source: §1.7.3 · p. 24
• Eleven iterations produce f(t) ≥ f11(t) > 1, proving the injectivity condition; symmetric bounds apply to g(t).Numerical computations suggest three iterations suffice, but eleven are used to minimize computer assistance.Source: §1.7.3 · p. 25, Figure 1.10 · p. 25

1.8.3 Optimality of Q at Gerver’s Sofa

• Q is globally concave on L, and Romik’s ODEs make its directional derivative at Gerver’s sofa nonpositive.The full proof uses 10 ODEs across five contact-pattern intervals; the ODEs balance differential side contributions so the derivative condition establishes optimality.Source: §1.8 · p. 25, §1.8.3 · pp. 31–32

1.8.1 Definition of Q

• For every monotone sofa, the constructed region R contains the sofa, has one core and two tails, and equals the sofa when it is Gerver’s sofa.The niche subset N′ is selected from three angular regions using the injectivity condition, and R is defined as K \ N′.Source: §1.8.1 · p. 26

1.8.2 Quadraticity of Q

• The cap, right-tail body, and left-tail body form a convex tuple (K, B, D) in L on which Q is a quadratic upper bound for sofa area.Representing R with K, B, and D makes its area amenable to quadratic analysis through the five boundary segments of N′.Source: §1.8.2 · pp. 27–28

1.8.3 Optimality of Q at Gerver’s Sofa

• |G| ≥ |S∗|, so Gerver’s sofa proves the main optimality theorem.Gerver’s region R equals G, while Q bounds the area of any maximum-area sofa satisfying injectivity; global concavity and local optimality at G yield the inequality.Source: §1.8.3 · p. 29

Monotone Sofas and Caps

• A maximum-area sofa is represented as a monotone sofa S = K \ N(K), where K is a cap containing its niche N(K).The monotone-sofa framework defines supporting hallways and shows that the cap–niche subtraction represents the sofa and that the niche lies within the cap.Source: Monotone Sofas and Caps · p. 33
• The sofa problem is recast as maximizing A_ω(K) := |K| − |N(K)| over caps K ∈ Kc, with rotation angle ω.This provides the optimization functional used for the sofa analysis.Source: Monotone Sofas and Caps · p. 33

2.1 Planar Convex Body

• Planar convex bodies are defined as nonempty compact convex subsets of R2, including points and line segments, with support lines, half-planes, widths, edges, and vertices specified.The section also introduces Hausdorff distance and surface area measure, which records edge lengths for convex bodies.Source: §2.1 · pp. 33–35

2.1. PLANAR CONVEX BODY

• Surface area measure σK assigns each direction the length of the corresponding edge, while for smooth boundaries its distribution function equals the radius of curvature.For polygons, σK is discrete; for smooth convex bodies, it varies continuously through curvature.Source: §2.1 · p. 35
• Vertices of a convex body vary continuously from the appropriate one-sided direction limits, yielding right- and left-continuity results for vK⁺(t).The proof confines nearby edges and their endpoints to a triangle of arbitrarily small diameter around vK⁺(t).Source: §2.1 · pp. 35–36

2.2 Supporting Hallway

• Supporting hallway LS(t) is the unique rotated hallway whose outer walls are tangent to S at normal angles t and t + π/2.Its parts can be expressed using supporting lines, half-planes, and the support function hS; replacing any containing hallway by LS(t) preserves containment.Source: §2.2 · pp. 37–38

2.3 Monotone Sofa

• Every moving sofa has a unique standard-position translation, up to horizontal translations when ω = π/2, contained in Pω.The standard position satisfies hS(ω) = hS(π/2) = 1.Source: §2.3 · pp. 38–39

Proof of Theorem 2.3.2

• For a standard-position sofa S, I(S) is a connected moving sofa with the same rotation angle and standard position, contains S, and therefore suffices for the moving sofa problem.The proof establishes S ⊆ I(S) ⊆ C(S), preserves support functions and supporting hallways on Jω, and uses these facts to prove connectedness and continuous motion.Source: §2.3 · p. 39, Proof of Theorem 2.3.2 · p. 40, §2.4 · p. 41

2.4 Cap and Niche

• The convex set C(S) associated with a standard-position moving sofa is a cap with rotation angle ω.Its cap characterization uses the prescribed support values and half-plane intersection structure.Source: §2.4 · p. 41
• The monotone sofa I(S) equals K \ N(K), where K = C(S) is the cap of S and N(K) is its niche.This gives the intersection construction a cap-and-niche representation.Source: §2.4 · pp. 41–42
• Any monotone sofa S with cap K equals K \ N(K), expressing it as its cap minus its niche.Source: §2.4 · p. 42
• The intersection operation enlarges moving sofas to monotone sofas while leaving monotone sofas unchanged.Source: §2.4 · p. 42
• For every moving sofa S′ in standard position, I(I(S′)) = I(S′), and S = I(S) holds exactly for monotone sofas.Source: §2.4 · p. 42

2.5. Cap Contains Niche

• For any cap K, its upper boundary δK is connected and forms the boundary of K in the relevant wedge region.The proof uses the connectedness of the parameter interval and the connected line segments defining the upper boundary.Source: §2.5 · p. 43
• For every interior angle t, both wedge gaps wK(t) and zK(t) are positive, ensuring the wedge geometry used to analyze the niche.The argument uses convexity and mirror symmetry to reduce the proof to one wedge gap.Source: §2.5 · p. 44
• If the inner corner xK(t) lies in K, then the wedge TK(t) is contained in K, with the proof divided into cases based on the origin’s position relative to the wedge boundaries.This geometric inclusion is used to control whether the niche lies inside the cap.Source: §2.5 · pp. 44–45
• A cap K is the cap of a monotone sofa if and only if it contains its niche N(K).The forward direction follows from representing the sofa as K \ N(K); the reverse direction reconstructs a moving sofa from K \ N(K).Source: §2.5 · p. 46
• Not every cap contains its niche: the long cap [0,100] × [0,1] has N(K) not contained in K and therefore cannot represent a monotone sofa at rotation angle π/2.This example shows why the cap-space maximization requires a later correspondence result rather than automatically applying to every cap.Source: §2.5 · p. 46
• The sofa area functional Aω(K) equals the area of the corresponding monotone sofa, allowing the moving sofa problem to be reformulated as maximizing Aω over cap space.This reformulation applies to caps of monotone sofas, while not every cap satisfies the niche-containment condition.Source: §2.5 · p. 46

Balanced Maximum Sofas and Caps

• A balanced maximum sofa Sω := Kω \ N(Kω) arises as the limit of maximum polygon caps and attains the maximum area among monotone sofas of rotation angle ω ∈ (0, π/2].The caps’ angle sets become denser in [0, ω] as the limit is taken.Source: Balanced Maximum Sofas and Caps · p. 47

3.1 Simple Nef Polygon

• Simple Nef polygons represent caps and niches through monotone Boolean combinations of half-planes with distinct boundaries, enabling side-balancing analysis.Theorem 3.1.2 measures the first-order area change when one defining half-plane is shifted, with a quadratic O(δ2) remainder.Source: §3.1 · pp. 47–48, §3.2 · p. 49

3.2 Polygon Cap and Niche

• For a finite angle set Θ with rotation angle ω, polygon caps are intersections of prescribed half-planes, while polygon niches and the area functional AΘ(K) = |K| −|NΘ(K)| define polygon sofas.Denser angle sets provide better approximations of the underlying cap.Source: §3.2 · pp. 49–50

3.2. POLYGON CAP AND NICHE

• The cap-and-niche construction depends only on support-function values at Θ ∪ (Θ + π/2), and polygon approximation preserves the niche inclusion NΘ(K) ⊆ N(K).For K′ = CΘ(K), the niche satisfies NΘ(K) = NΘ(K′).Source: §3.2 · p. 50

3.3 Extensions of Polygon Cap Space

• The framework extends translated polygon caps to HΘ, the space of functions h : Θ⋄ → R, while preserving cap, niche, area, and translation-equivariance properties.The extended cap and niche remain simple Nef polygons, and for translated caps the niche translates while AΘ remains unchanged.Source: §3.3 · pp. 51–52
• K+ is contained in Ph+ and has width 1 in the directions uω and uπ/2, so it is circumscribed in Ph+ and satisfies hK+(ω) = h+(ω) and hK+(π/2) = h+(π/2).Source: §3.3 · p. 53
• The pointwise inequality hK+(t) ≤ h+(t) on Θ⋄ implies NΘ(hK+) ⊆ NΘ(h+).Source: §3.3 · p. 53
• Applying Proposition 3.3.4 and Definition 3.3.4 gives CΘ(hK+) = K+ = CΘ(h+).The final equality uses Definition 3.3.4.Source: §3.3 · p. 53

Symmetry

• Reflecting a maximum polygon cap across the symmetry axis produces a maximum polygon cap with angle set ω − Θ.The functional A_Θ and the distinguished point o_ω are preserved by the reflection.Source: §3.4 · p. 53

Boundedness

• Any cap with positive A_Θ has uniformly bounded width along u_0, including when ω = π/2.For ω = π/2, a sufficiently wide cap makes the quadratic lower bound on niche area exceed its area, forcing A_Θ(K) ≤ 0.Source: §3.4 · p. 53

Existence

• A maximum polygon cap K_Θ exists for every angle set Θ.Existence follows by restricting to a nonempty compact class of caps, using bounded width and continuity of A_Θ under Hausdorff convergence.Source: §3.4 · pp. 53–54

Balancedness

• Every maximum polygon cap is balanced, meaning σ_K(t) = τ_K(t) for every t ∈ Θ⋄.If σ_K(t) > τ_K(t) for some angle, a small support-function increase yields a translated cap with larger objective, contradicting maximality.Source: §3.4 · pp. 56–58

Niche containment

• Any maximum polygon cap K with angle set Θ contains its polygon niche N_Θ(K).The proof reduces containment to showing every vertex of the associated polyline p_K lies in K, using balancedness and mirror symmetry.Source: §3.4 · p. 58

• Balancedness is imposed through τ_K(t) = σ_K(t) while following the upper boundary δ_K from A−.Source: §3.4 · p. 58
• The resulting relation K(s) · u_s implies p ∈ H_K(s).Source: §3.4 · p. 58

3.5 BALANCED MAXIMUM SOFA

• The mirror reflection of any balanced maximum cap is also a balanced maximum cap.This symmetry is established by applying the reflection property to the convergent sequence of maximum polygon caps.Source: §3.5 · p. 59

3.5 Balanced Maximum Sofa

• For every ω ∈ (0, π/2], a balanced maximum cap Kω exists as the Hausdorff limit of maximum polygon caps with increasingly dense uniform angle sets.The approximating caps use angle sets Θω,2^i and converge in Hausdorff distance.Source: §3.5 · p. 59

3.5 BALANCED MAXIMUM SOFA

• The limit cap satisfies N(Kω) ⊆ Kω, so Sω is connected and forms a monotone sofa.This containment follows by passing the polygon-niche containment through Hausdorff convergence.Source: §3.5 · p. 59
• The polygonal area values converge to the limiting area Aω(Kω), and maximality of the approximating caps proves that Kω attains the supremum of Aω.The proof controls both cap areas and niche areas under convergence.Source: §3.5 · pp. 59–60

3.5 Balanced Maximum Sofa

• A balanced maximum sofa Sω exists and attains the maximum area among all moving sofas with rotation angle ω ∈ (0, π/2].It is obtained from a balanced maximum cap Kω as Sω = Kω \ N(Kω).Source: §3.5 · p. 60

4.1 Horizontal Side Lengths

• For ω < π/2, every balanced maximum cap satisfies σK(π/2) ≥ w◦K, a horizontal-side comparison used in proving that a maximum-area sofa rotates through π/2.The comparison follows from polygonal approximations, weak convergence of surface-area measures, and balancedness.Source: §4.1 · p. 62

4.2 Right Rotation Angle

• The right-rotation analysis defines the geometry of Pω and Rω,d, choosing dω,min as 1.25 or 1.1 and proving area(Rω,dω,min) < 2.2.For smaller angles the region area is computed explicitly, while convexity estimates support the required endpoint checks.Source: §4.2 · pp. 62–64

4.2. RIGHT ROTATION ANGLE

• Theorem 4.2.5 shows that any balanced maximum cap with rotation angle ω and area at least 2.2 contains a suitable quadrant configuration, yielding the main step toward ω = π/2.The proof bounds an auxiliary region below 2.2, then uses support heights, convexity, and cone geometry to place the three specified points in the quadrant closure.Source: §4.2 · pp. 63–65
• Theorem 4.2.5 implies that Sω can be transformed into a movement with rotation angle π/2, so |Sω| ≤ |Sπ/2| and Theorem 1.5.2 follows.The argument places the rotated sofa inside the required hallway while preserving its area and then follows the original movement.Source: §4.2 · pp. 65–66

5.1 Lebesgue–Stieltjes Measure

• The Lebesgue–Stieltjes measure df is defined for right-continuous functions of bounded variation, with linearity and a product rule when one factor is continuous.The section also characterizes absolute continuity by df = r dt, with f′(t) = r(t) almost everywhere.Source: §5.1 · pp. 67–68

5.2 Differential Gauss–Minkowski Theorem

• The differential surface-area framework establishes bounded variation for v^+_K(t) and states that dv^+_K(t) is evaluated through the surface-area measure σ_K.Monotonicity on angular subintervals supplies bounded variation across the full parameter domain.Source: §5.2 · p. 69
• Theorem 5.2.2 gives an equality between endpoint vertex changes and integrals of normal directions against the surface area measure of any planar convex body.The result applies to arbitrary a,b with a < b ≤ a + 2π and distinguishes the body-dependent vertex v+_K(t) from the direction v_t.Source: §5.2 · p. 69
• The proof first verifies the identity for polygons by expressing the surface area measure as point masses on proper edges, whose vector contributions form a telescoping sum.For general convex bodies, polygonal approximations converge in Hausdorff distance, while the associated restricted measures converge weakly to the limiting surface area measure.Source: §5.2 · pp. 69–70
• The stated equality does not generally hold at the left endpoint {a}, because the Lebesgue–Stieltjes measure is defined to vanish there.Source: §5.2 · p. 69
• Integrating the differential theorem over the full circle recovers the classical zero-vector condition for the surface area measure and, conversely, reconstructs a convex body from a measure satisfying that condition.The reconstruction uses the convex hull of partial integrals of v_t against the measure, whose points are in convex position and recover the corresponding vertex functions.Source: §5.2 · p. 70

Injectivity Condition

• The polygonal inequality is designed to pass to the limit as n →∞, producing the continuous differential inequality for balanced maximum caps.Maximum polygon caps converge in Hausdorff distance to a balanced maximum cap, whose surface-area measure is correspondingly bounded through the limiting argument.Source: Injectivity Condition · p. 71, §6.3 · pp. 74–75

6.1 Statement

• Any balanced maximum cap satisfies the injectivity condition, and Gerver’s sofa cap satisfies it as a consequence.The condition requires measurable functions r_K and s_K, a continuously differentiable inner corner x_K, and two directional derivative inequalities.Source: §6.1 · pp. 71–72

6.2 Arm Lengths

• Arm lengths f_K and g_K express the derivatives of the cap’s outer and inner corners, making the injectivity inequalities equivalent to f_K^±(t) > 1.The derivative formulas are established using one-sided derivatives, with corresponding left- and right-continuity properties for the arm-length functions.Source: §6.2 · pp. 72–74

6.3 Inequality on Maximum Polygon Caps

• For maximum polygon caps with uniform angle sets and step size δ = (π/2)/n, balancedness yields an upper bound on each side-length measure σ_K(t).The argument first bounds cap diameters and arm lengths, then uses balancedness to relate σ_K(t) to the corresponding polygon-niche side length.Source: §6.3 · pp. 75, 77

6.3. INEQUALITY ON MAXIMUM POLYGON CAPS

• The boundary segment of bK(t) ∩ S has length max(0, 2 tan(δ/2) − σK(t)), obtained by comparing the intersections at t + δ and t − δ.A unique β relates the two intersections, with β = 2 tan(δ/2) − σK(t); the segment is empty for β < 0 and has length β otherwise.Source: §6.3 · p. 78
• H1(X \ R) ≤ max(0, δ − s) + O(δ^2), providing the key estimate for the portion outside R.This follows from the sharper bound H1(X \ R) ≤ max(0, 2 tan(δ/2) − σK(t)).Source: §6.3 · p. 78
• Using s = νK(t) = H1(X), the proof considers separately the cases s ≤ δ and s > δ, and both cases establish the claimed inequality.The two case bounds are obtained by adding the estimates in Equations (6.1) and (6.2).Source: §6.3 · p. 78

6.4 Inequality on Balanced Maximum Caps

• Adjacent polygon angles control the approximation error because the corresponding trajectory arc has length at most 5δ.The trajectory lies on a circle of diameter at most 5 and has central angle 2δ.Source: §6.4 · p. 78

6.4. INEQUALITY ON BALANCED MAXIMUM CAPS

• σK ≤ k0(g+K(t)) dt holds on [0, π/2) for every balanced maximum cap K.Source: §6.4 · pp. 79–80

6.4 Inequality on Balanced Maximum Caps

• The continuous inequality is obtained by approximating K with maximum polygon caps whose dyadic angle sets become dense as n →∞.The proof passes from polygon inequalities to interval inequalities and then uses measure convergence and dominated convergence.Source: §6.4 · pp. 79–80

6.4. INEQUALITY ON BALANCED MAXIMUM CAPS

• The limit argument handles both open intervals and intervals of the form [0,b), yielding σK(I) ≤ µK(I) in each case.Source: §6.4 · p. 80
• By the Radon–Nikodym theorem and mirror symmetry, every balanced maximum cap satisfies condition (1) of Definition 6.1.2 across the relevant angle ranges.The condition also gives common values for the associated endpoint quantities and motivates the notation AK(t) and fK(t).Source: §6.4 · p. 80

6.5 Bounding Arm Lengths

• Balanced maximum caps have absolutely continuous f_K with f′_K(t) ≥ m_0(g_K(t)) almost everywhere, enabling iterative lower bounds for both arm-length functions.The derivative follows from the cap measure representation and the bound r_K(t) ≤ k_0(g_K(t)).Source: §6.5 · p. 81

6.5. BOUNDING ARM LENGTHS

• The inequalities f_K, g_K > 1, together with the corner derivative formula, establish the injectivity condition and complete the proof of Theorem 6.1.1 for balanced maximum caps.Conditions (1) and (2) come from earlier results, while condition (3) follows from Proposition 6.4.6 and Theorem 6.5.6.Source: §6.5 · pp. 81, 83

6.5 Bounding Arm Lengths

• The iterative functions f_0 = 0 and f_{n+1} = max(f_n, Ff_n) provide universal lower bounds f_K(t) ≥ f_n(t) and g_K(t) ≥ f_n(π/2 − t).The operator F is monotone, and the bounds propagate inductively using mirror symmetry.Source: §6.5 · pp. 81–83

6.5. BOUNDING ARM LENGTHS

• f_11(x) > 1 for x ∈ (0, 1], and for every balanced maximum cap K, f_K(t) > 1 on t ∈ (0, π/2] while g_K(t) > 1 on t ∈ [0, π/2).Source: §6.5 · p. 83

Convex Domain and Convex Curves

• Chapter 7 introduces convex domains, linear and quadratic functionals, curve-area functionals, convex boundary segments, and Mamikon’s theorem as preparation for Chapter 8.The overview assigns these topics to Sections 7.1–7.4.Source: Convex Domain and Convex Curves · p. 84

7.1 Convex Domain

• For a concave quadratic functional on a convex domain, a point maximizes the functional exactly when its directional derivative is a nonpositive convex-linear functional.The framework applies to planar convex bodies under Minkowski barycentric operations, with area identified as a quadratic functional.Source: §7.1 · pp. 85–86

7.2. CURVE AREA FUNCTIONAL

• J(x) is quadratic on continuous bounded-variation maps, and its value J(Γ) is independent of the parametrization for any oriented Jordan arc or curve.For nonclosed arcs, J measures the signed area bounded by the arc and the two segments joining its endpoints to the origin.Source: §7.2 · pp. 87–88
• For a rectifiable counterclockwise Jordan curve, J(x) equals the area enclosed by the curve.This follows by applying Green’s theorem to the vector field (P, Q) = (−y, x).Source: §7.2 · p. 88
• For an oriented line segment from p to q, J(p, q) = (p × q)/2; if the segment lies on l(t, h) with displacement d v_t, then J(p, q) = hd/2.If p, q, and the origin are collinear, this value is zero.Source: §7.2 · p. 88
• The curve area functional is additive when an oriented Jordan arc or curve is formed by concatenating Jordan arcs in order.The concatenated arcs meet at successive endpoints, allowing J(Γ) to be computed from the component contributions.Source: §7.2 · p. 88
• A concatenated Jordan curve contained in a half-plane is counterclockwise when one component is a positively oriented line segment along the boundary line.The segment must have positive length and lie in the positive v_t direction.Source: §7.2 · p. 88

7.3 Convex Curve

• For a convex body and support directions a < b < a + π, the constructed set ua,bK is a rectifiable oriented Jordan arc joining the designated endpoint vertices.Source: §7.3 · pp. 89–90
• The construction remains valid in degenerate cases, including when the relevant convex body collapses to a line segment or a single point.Source: §7.3 · pp. 89–90
• The curve-area functional of ua,bK is quadratic in K.Source: §7.3 · p. 90

7.3. CONVEX CURVE

• The arc construction is compatible with concatenation across intermediate directions, supporting decomposition of longer convex curve segments.Source: §7.3 · p. 91
• The auxiliary curve formed from ua,bK and two connecting segments is counterclockwise, lies inside HK(a) ∩ HK(b), and avoids the common intersection of the intermediate half-planes.Source: §7.3 · p. 91

7.4 Mamikon’s Theorem

• Mamikon’s theorem is generalized to arbitrary convex bodies with non-differentiable boundaries [Mna97].The generalized result expresses the Mamikon-region area through the integral of the squared tangent-segment length.Source: §7.4 · p. 92

7.4. MAMIKON’S THEOREM

• Under convex-linear dependence of zK on K, the Mamikon-region functional MK(a, b; zK) is quadratic and convex in K.The proof uses the generalized area identity and the convex-linearity of αK.Source: §7.4 · p. 93

Chapter 8

• Chapter 8 uses Mamikon’s theorem to establish concavity of the upper-bound functional Q after defining its domain, proving its upper-bound property, and translating Gerver’s local optimality conditions.The chapter’s later steps calculate the directional derivative of Q at the tuple arising from Gerver’s sofa.Source: Chapter 8 · p. 94

8.1 Domain of Q

• The domain Ki is convex, contains all balanced maximum caps and Gerver’s cap C(G), and is extended to a convex tuple domain L for Q.Ki consists of injective caps with area at least 2.2; L imposes containment, support-function inequalities, and endpoint equalities on (K, B, D).Source: §8.1 · pp. 94–95

8.1. DOMAIN OF Q

• Lemma 8.1.7 bounds the support functions by h_K(t) + h_B(π + t) ≤ 1 for t ∈ [φ_R, π/2] and h_K(π/2 + t) + h_D(3π/2 + t) ≤ 1 for t ∈ [0, φ_L].Equality holds at t = φ_R, π/2 for the first inequality and at t = 0, φ_R for the second, yielding the stated boundary-line identities for B and D.Source: §8.1 · pp. 97–98
• Theorem 8.1.8 shows that for every K ∈ K_i, the triple (K, B_K, D_K) belongs to L.The result follows by verifying Definition 8.1.3: one condition is immediate from the definitions, while the remaining conditions are supplied by Lemma 8.1.7.Source: §8.1 · p. 98
• The injection K_i → L is not convex-linear, despite both K_i and L being convex domains.Its lower-left component B_K is the Aleksandrov body, or Wulff shape, associated with the relevant support-function construction.Source: §8.1 · p. 98
• The injection ‘irons out’ the nonquadratic sofa area functional A : K_i → R into the quadratic functional Q : L → R.This makes the map K ↦ (K, B_K, D_K) important for analyzing the area functional.Source: §8.1 · p. 98

8.2 Definition of Q

• Q is a quadratic functional on the domain L of convex-body tuples (K, B, D).The construction decomposes the niche into parts associated with K, B, and D and expresses their areas through quadratic terms.Source: §8.2 · pp. 98–99
• The upper-bound construction uses Jordan curves and enclosed regions to represent niche-area contributions geometrically.The proof builds closed curves from convex boundary arcs and line segments, then applies area identities to the enclosed regions.Source: §8.2 · pp. 99–100
• For every K in Ki, the sofa area satisfies A(K) ≤ Q(K, B_K, D_K).The bound is obtained by combining inequalities controlling the right, middle, and left parts of the niche.Source: §8.2 · pp. 99–100

8.3 Concavity of Q

• Mamikon regions make Q concave because their areas are convex while Q plus the relevant region area is linear in K.The proof introduces named Mamikon regions whose functionals are convex and quadratic, then combines them with Q.Source: §8.3 · pp. 101–104
• Theorem 8.3.8 establishes that Q : L → R is concave.This follows by combining the quadratic-concave decomposition with the supporting-line and Mamikon-area lemmas.Source: §8.3 · pp. 103–104

Boundary Curves of G

• Gerver’s sofa G is a monotone sofa whose boundary is parametrized by five oriented curves A, B, C, D, and x across five intervals.The intervals partition [0, π/2], and x is solved piecewise through Romik’s ODE system.Source: Boundary Curves of G · p. 105
• The niche boundary is formed by reversed B, restricted x, and reversed D, while supporting-hallway contacts and tangent directions constrain B and D.These conditions encode how the boundary curves meet hallway walls and move relative to the unit directions u_t and v_t.Source: Boundary Curves of G · pp. 105–106

ODEs of Boundary Curves of G

• Each interval has a fixed contact-point set Γ(t), and balancing differential side lengths yields the corresponding ODEs for A, B, C, D, and x.The ODEs are Romik’s equations recast for the five interval-specific contact configurations.Source: ODEs of Boundary Curves of G · pp. 106–107

Left, Middle, and Right Parts of G

• The construction also establishes matching tail, support-line, and support-function relations for the left and right parts of G, including h_K(π/2+t)+h_D(3π/2+t)=1.These identities are used to relate the cap K to the auxiliary convex bodies B and D.Source: Left, Middle, and Right Parts of G · p. 108

Surface Area Measures of G

• For the stated intervals, the surface-area measure satisfies σK=0 on J1, σK=ιK on J2∪J3, σK=˘σB+ιK on J4, σK=˘σB on J5, and σK=˘σD on J6.Here, ιK(dt)=iK(t)dt and ιK({0})=0.Source: Surface Area Measures of G · p. 109

8.5 Directional Derivative of Q

• The area, mixed-volume, point-pair, and curve-area terms are treated as quadratic functionals so their directional derivatives can be combined into a formula for Q.The curve-area contribution of the restricted inner-corner curve x_K is handled separately on I=[φ_R,φ_L].Source: §8.5 · pp. 110–112

8.5. DIRECTIONAL DERIVATIVE OF Q

• The directional derivative DQ(K,B,D;K*,B*,D*) is non-positive for every feasible direction from Gerver’s triple.The proof splits the derivative across the interval measures and shows each contribution is zero or has the required sign.Source: §8.5 · p. 113
• On intervals J_1, J_2∪J_3, and J_8∪J_9, the measure identities make the corresponding derivative contributions zero.The remaining B- and D-supported intervals are controlled using support-function inequalities and nonnegative reflected measures.Source: §8.5 · p. 113

8.5 Directional Derivative of Q

• The triple (K,B_K,D_K) from Gerver’s sofa attains the maximum of Q over L.This follows from the non-positive directional derivative and the concavity of Q.Source: §8.5 · p. 113
• For any maximum-area sofa S* with cap K*, the inequalities |S*|≤Q(K*,B_K*,D_K*)≤Q(K,B,D)=|G| prove that G attains the maximum area.The upper bound is exact at Gerver’s sofa, completing the global optimality argument.Source: §8.5 · p. 114

APPENDIX A. TABLE OF SYMBOLS

• It lists moving-sofa and cap spaces, including S, G, C(S), K^c, K_i, and the space L of triples (K,B,D).The same entries define the niche, wedges, angle sets, and boundary vertices used throughout the proof.Source: Table of Symbols · p. 115, Appendix A · pp. 116–117

Table of Symbols

• The appendix defines core geometric notation, including caps K, support functions h_K, supporting lines l_K, hallways L, and their walls and corners.It also records the unit vectors u_t and v_t and the rotation operator R_t.Source: Table of Symbols · p. 115

APPENDIX A. TABLE OF SYMBOLS

• The optimization notation includes surface-area measures σ_K, directional derivatives D_f, the upper-bound functional Q, curve-area functionals J, and the five intervals and curves defining G.The constants φ_R and φ_L and the partition angles t_0,…,t_5 specify Gerver’s parameterization.Source: Appendix A · pp. 116–117