Plücker bodies and linear isometry recovery

Scope

Boundary integrals and a common orientation compare the Plücker bodies of the two norms. Support slices, generator matching and maximal-minor factorization give approximate linear recovery, then the limiting volume argument gives an ambient linear isometry. This existence conclusion does not assert extension of a specified sphere isometry.

14 direct Cards + 57 reused prerequisites = 71 unique Cards. This count is a selected Card closure, not a source-declaration count.

Route reading PDF · Preserved source exploration

Cards in this route

Key result: The unit-sphere chord metric determines the ambient normed space
Key result: All Plücker bodies determine the norm up to linear isometry

Read this route with prerequisites

Reference index: direct Cards and reused prerequisites

Direct references: SR093 — The symmetric convex body of contraction minors · SR098 — Compactness of the signed generator set · SR345 — The set average of derivative generators · SR347 — The average of contractive derivative generators lies in the body · SR022 — A seminorm increment bound passes to every derivative direction · SR354 — One boundary extension with all five analytic properties · SR363 — A common orientation sign for the radial derivative average · SR350 — A target contraction generator lies in the source body · SR353 — Sphere isometries preserve the Plücker body · SR302 — The chosen support slice consists of almost-isometric generators · SR305 — Equal Plücker bodies yield matching generators with an almost-isometric representative · SR365 — Almost-contractive linear recovery with exact volume scaling · SR359 — All Plücker bodies determine the norm up to linear isometry · SR383 — The unit-sphere chord metric determines the ambient normed space

Reused prerequisites: SR328 — The satellite polynomial is linear in the maximal-minor vector · SR081 — Successive maximum refinement by an ordered list of functionals · SR170 — Equality of seminorm balls under a linear bijection preserves the seminorms · SR168 — A compact proper subset of a seminorm ball has smaller Haar measure · SR372 — Finite-dimensionality transfers across a sphere isometry · SR406 — The family of maximal minors in increasing row order · SR392 — The inverse sphere isometry gives the inverse radial map · SR425 — Cramer’s rule for coordinates of a row · SR387 — Radial extension of an isometry between unit spheres · SR374 — Isometric unit spheres force equal finite dimensions · SR019 — The signed Jacobian integral is a sign times target volume · SR455 — Determinant integrals under strong convergence · SR032 — Strong local convergence of mollified derivatives · SR017 — Almost-everywhere constancy of the sign on a preconnected target · SR230 — A common inverse estimate from row Cramer · SR444 — Gluing one almost-everywhere constant over a countable cover · SR264 — A seminorm with two-sided bounds against a reference norm · SR050 — The divergence-free cofactor identity · SR235 — A weighted determinant polynomial with satellites · SR157 — Zero weak gradient gives zero derivatives of local mollifications · SR099 — Divergence as the trace of a derivative · SR006 — A positive lower metric bound passes to the derivative · SR118 — Global almost-everywhere constancy on a preconnected domain · SR325 — A strict lower bound on a norm’s unit sphere · SR065 — The convex hull of a compact set in a finite real coordinate space · SR120 — Vanishing weak gradient tested by divergence · SR263 — A finite absolute-maximizing family almost norms the unit sphere · SR086 — A common generator selected from equal convex hulls · SR064 — Smooth compact perturbations preserve the determinant integral difference · SR253 — Absolute maximization forces a near-maximal base · SR272 — Continuous linear maps bounded by a seminorm · SR117 — Local almost-everywhere constancy from the weak equation · SR297 — A determinant difference bound in the sup operator norm · SR155 — Differentiating a local mollification through its kernel · SR015 — Signed transfer from the absolute Jacobian formula · SR356 — A volume-normalized linear contraction obtained by a limit · SR449 — Identifying constants on an open overlap · SR464 — Maximal-minor integral differences for Lipschitz perturbations · SR465 — Boundary agreement determines maximal-minor integrals · SR439 — Maximal minors scaled by the reference ball volume · SR258 — One fixed satellite family almost norms every detected unit vector · SR020 — Weak Piola identity for a compactly supported test field · SR394 — The radial extension is 3-Lipschitz · SR216 — The set of near-maximal dual frames · SR036 — A cofactor row by row replacement · SR061 — A component flux gives a determinant difference · SR058 — Zero integral of a compactly supported divergence · SR021 — The transported Jacobian sign has zero weak gradient · SR025 — Mollification by a normalized smooth kernel · SR190 — An attained absolute determinant maximum · SR207 — Positivity of the determinant maximum · SR358 — The recovered contraction maps one unit ball onto the other · SR077 — A support face is the convex hull of the maximizing generators · SR255 — Each satellite norms its coefficient preimage in absolute value · SR001 — Inverse maps on open sets with global metric bounds · SR452 — From local to global almost-everywhere constancy · SR257 — A nearby norming point gives an almost-norming evaluation

Dependency-first reading route

Read selected Card prerequisites before their uses. Levels are recomputed from the selected reachability-preserving projection; omitted source helpers remain traceable in the source exploration.

Search Cards Route 71 Cards

Level 0 (21 Cards)

Level 0

A seminorm increment bound passes to every derivative direction

Takes a difference-quotient limit without imposing a positive lower bound.

MathlibAnnex.FDeriv.norm_apply_le_seminorm_of_lipschitz

Immediate Card prerequisites: None in this selected scope

Used by in this scope: One boundary extension with all five analytic properties

Scope

Cards in this route

Dependency-first reading route

Level 0 (21 Cards)

A seminorm increment bound passes to every derivative direction

A weighted determinant polynomial with satellites

The family of maximal minors in increasing row order

Gluing one almost-everywhere constant over a countable cover

A determinant difference bound in the sup operator norm

Successive maximum refinement by an ordered list of functionals

Mollification by a normalized smooth kernel

Divergence as the trace of a derivative

Cramer’s rule for coordinates of a row

Radial extension of an isometry between unit spheres

A compact proper subset of a seminorm ball has smaller Haar measure

A cofactor row by row replacement

A support face is the convex hull of the maximizing generators

Zero integral of a compactly supported divergence

The convex hull of a compact set in a finite real coordinate space

A seminorm with two-sided bounds against a reference norm

Inverse maps on open sets with global metric bounds

An attained absolute determinant maximum

Equality of seminorm balls under a linear bijection preserves the seminorms

Identifying constants on an open overlap

Differentiating a local mollification through its kernel

Level 1 (18 Cards)

Absolute maximization forces a near-maximal base

Positivity of the determinant maximum

Strong local convergence of mollified derivatives

The divergence-free cofactor identity

Continuous linear maps bounded by a seminorm

A strict lower bound on a norm’s unit sphere

Vanishing weak gradient tested by divergence

The satellite polynomial is linear in the maximal-minor vector

Maximal minors scaled by the reference ball volume

A positive lower metric bound passes to the derivative

The set of near-maximal dual frames

From local to global almost-everywhere constancy

The radial extension is 3-Lipschitz

The inverse sphere isometry gives the inverse radial map

A nearby norming point gives an almost-norming evaluation

The recovered contraction maps one unit ball onto the other

Determinant integrals under strong convergence

A common generator selected from equal convex hulls

Level 2 (10 Cards)

Compactness of the signed generator set

The set average of derivative generators

The symmetric convex body of contraction minors

Isometric unit spheres force equal finite dimensions

A common inverse estimate from row Cramer

A component flux gives a determinant difference

Zero weak gradient gives zero derivatives of local mollifications

Signed transfer from the absolute Jacobian formula

Finite-dimensionality transfers across a sphere isometry

Each satellite norms its coefficient preimage in absolute value

Level 3 (4 Cards)

The average of contractive derivative generators lies in the body

Smooth compact perturbations preserve the determinant integral difference

Local almost-everywhere constancy from the weak equation

One fixed satellite family almost norms every detected unit vector

Level 4 (5 Cards)

One boundary extension with all five analytic properties

The chosen support slice consists of almost-isometric generators

A finite absolute-maximizing family almost norms the unit sphere

Global almost-everywhere constancy on a preconnected domain

Maximal-minor integral differences for Lipschitz perturbations

Level 5 (4 Cards)

Almost-contractive linear recovery with exact volume scaling

Equal Plücker bodies yield matching generators with an almost-isometric representative

Boundary agreement determines maximal-minor integrals

Weak Piola identity for a compactly supported test field

Level 6 (2 Cards)

A volume-normalized linear contraction obtained by a limit

The transported Jacobian sign has zero weak gradient

Level 7 (2 Cards)

All Plücker bodies determine the norm up to linear isometry

Almost-everywhere constancy of the sign on a preconnected target

Level 8 (1 Card)

The signed Jacobian integral is a sign times target volume

Level 9 (1 Card)