Convex packets, extremal norms, and ellipsoidal compression

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The geometry appropriate to a packet depends on its task. For exact invertible isometries, arbitrary Finsler norms do not enlarge the completion class. For one-sided semigroup control, they can be strictly stronger than quadratic metrics.

Import the implementation explicitly from komorebi_math.nonlinear.convex_packets. Related contracts are described in metric packets and distortion cocycles.

1. Convex geometric packets

A finite-dimensional norm is equivalent to a centrally symmetric convex body K containing the origin in its interior. The unit ball is the geometric packet. For a linear map A,

One explicit family provided by the API is

where F is invertible. Its unit ball is a parallelotope. Under a coordinate change x'=Gx, the packet transports as F'=FG^{-1}, so vector and operator norms are coordinate-covariant.

2. Exact invertible isometry: no Finsler enlargement

Let Gamma <= GL(V) preserve some norm exactly:

Then every g is a linear isometry of a bounded convex body. Hence Gamma is bounded, its closure is compact, and averaging any Euclidean inner product over the compact closure gives a positive-definite invariant inner product. Thus:

Finite-dimensional isometry theorem. An invertible linear group admits an invariant norm if and only if it admits an invariant positive-definite quadratic norm.

The same applies to exact conformal transport after removing the scalar factor from each generator. Therefore replacing inner products with arbitrary Finsler norms does not remove the holonomy metric-completion obstruction.

3. Forward semigroups are different

For a finite family A={A_1,...,A_m} and r>0, let S_r be the semigroup generated by A_i/r.

Product-bounded completion theorem. There exists a norm with ||A_i|| <= r for every generator if and only if S_r is product-bounded.

If the norm exists, all normalized products are nonexpansive and hence bounded. Conversely, if S_r is bounded in a reference Euclidean norm, define

This is a finite norm and each A_i/r is nonexpansive in it.

The classical Rota--Strang variational formula gives

where rho is the joint spectral radius. When this intrinsic growth rate is positive, normalization by it gives two cases for arbitrary norm packets:

1. convex completion: the normalized semigroup is product-bounded and an extremal norm is attained;
2. convex escape: the infimum is one but no extremal norm is attained.

The unipotent Jordan block is an escape example: its joint spectral radius is one, but its powers grow linearly, so no norm makes it nonexpansive. Diagonal coordinate packets can nevertheless make its one-step bound approach one while the packet condition number diverges.

For compact irreducible matrix families, Barabanov norms provide an important attainment mechanism. These facts are established joint-spectral-radius machinery.

4. Sharp convex-versus-quadratic separation

Consider

In the infinity norm,

so

Thus the square is an exact forward-invariant convex packet and the joint spectral radius is one.

Now let

Both maps are rank one. Their squared quadratic operator norms are

If both were strictly less than 2, adding the resulting inequalities would force

which is impossible. Hence

for every positive metric. Q=I attains equality, so

while the optimal arbitrary norm bound is exactly one.

This example is the standard Ando--Shih separation used in the joint spectral radius and Lyapunov literature. It shows that a quadratic gap need not be an obstruction in a richer geometry class.

5. John ellipsoid compression

For a symmetric convex body K in R^n, John's theorem gives an ellipsoid E with

If A K subseteq r K, then

so the associated quadratic norm satisfies

Consequently the best ellipsoidal joint bound obeys

This is the classical ellipsoidal approximation guarantee. The two-dimensional pair above attains the factor sqrt(2), so the generic dimension bound is sharp already for the two-dimensional example.

For the implemented parallelotope packet ||Fx||_inf, the John ellipsoid is explicit:

The implementation therefore exposes both the exact convex operator norm and its compressed quadratic counterpart.

6. Geometry classes and obstruction hierarchy

A positive quadratic gap must be labeled by its admissible geometry class. It need not be an obstruction in a richer convex packet category.

For forward dynamics, a useful hierarchy is

The intrinsic growth obstruction is the joint spectral radius. Above that floor, failure to attain the bound is an escape/compactness issue; extra gap in a restricted packet class is a model-class obstruction.

For invertible exact isometry/conformal holonomy, however, the zero-obstruction class does not enlarge: invariant Finsler structure can be averaged back to an invariant quadratic structure.

This task dependence is essential. "Metric is too restrictive" is true for one-sided semigroup certification and false for exact finite-dimensional group isometry.

7. Verification boundary

The regression suite checks:

• coordinate covariance of parallelotope packets;
• explicit John metric compression;
• exact infinity-norm nonexpansivity of the Ando--Shih pair;
• the sharp sqrt(2) quadratic lower bound under randomized positive-metric pressure tests;
• convex escape of the unipotent family.

Additional deterministic trials compare parallelotope bounds with their John-ellipsoid compression. These finite checks test the implementation over their sampled inputs. The mathematical statements use classical convex geometry, joint spectral radius, and switched-system theory.