The Grassmann distance complexity

Andrea Rosana; Antonio Lerario

REVIEW 3 major objections 5 minor 22 references

The Grassmann distance complexity

T0 review · 3 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Nearest-point search on Grassmannians has finite generic complexity, and Schubert varieties are solved by Eckart–Young.

desk verdict A new complexity measure for the Grassmannian with a clean nonlinear Eckart–Young theorem, but the finiteness theorem rests on a plausible, unproved definability assertion. read the letter →

arxiv 2411.16589 v1 pith:E3TUB4K4 submitted 2024-11-25 math.DG math.AG

Antonio Lerario , Andrea Rosana This is my paper · ORCID

classification math.DGmath.AG MSC 53C2214M1532B20

keywords GrassmanndistancecomplexityEuclideandegreesubanalyticsetso-minimalgeometryLipschitzcriticalpointtheorySchubertvarietiesEckart-YoungtheoremPfaffianfunctions

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper introduces the Grassmann Distance Complexity (GDC) of a subanalytic submanifold of a Grassmannian: the maximum, over generic external $k$-planes $L$, of the number of critical points of the intrinsic Riemannian distance from $L$ that lie outside $L$'s cut locus. The central claim is that this number is finite and bounded by a constant $C_X$ depending only on the submanifold $X$ (Theorem 42). Because the Grassmann distance is neither smooth nor semialgebraic, the proof works with Clarke's subdifferential for Lipschitz functions and with o-minimal geometry, a tame logical framework in which principal angles are definable. For real algebraic hypersurfaces the bound is made explicit in the degree via Pfaffian function theory. The paper also proves a nonlinear Eckart–Young theorem: for simple Schubert varieties $\Omega_s$, the unique closest point to a generic $L$ is obtained by zeroing the $s$ smallest principal angles, with distance $(\theta_1(L)^2+\cdots+\theta_s(L)^2)^{1/2}$.

What carries the argument

The load-bearing object is the Clarke subdifferential $\partial_S\delta_L$ of the Lipschitz distance function, together with the description of the cut locus as the Schubert variety $\mathrm{cut}(L)=\{E:\theta_k(L,E)=\pi/2\}$, stratified by the number of principal angles equal to $\pi/2$. At a point in the $j$-th stratum the subdifferential is a convex set linearly isomorphic to the convex hull of the orthogonal group $O(j)$, of dimension $j^2$. Finiteness of critical points away from the cut locus comes from the normal exponential map restricted to the normal bundle $NX$, whose generic fibers are discrete by Sard's theorem; o-minimal definability of the principal-angle functions then makes the fiber count uniformly bounded. The Pfaffian bound for hypersurfaces encodes the singular-value/exponential description of the distance in a Pfaffian chain and controls its Betti numbers by known estimates.

What would settle it

For a concrete smooth subanalytic $X$, write the critical-point relation explicitly and check whether it is definable in an o-minimal structure; a single $X$ for which this relation is not definable, or a sequence of generic $L_m$ with an unbounded number of critical points outside $\mathrm{cut}(L_m)$, would refute Theorem 42. A cheaper numerical check for a Schubert variety like $\Omega_s\subset G(3,6)$ is to count, for random $L$, all critical points of $\delta_L|_{\Omega_s}$ outside the cut locus and see whether the count exceeds $\binom{k}{s}$.

Watch

Extended reading notes

Core claim

The central discovery is that the nearest-point problem on a Grassmannian, despite the non-smoothness and non-semialgebraicity of the metric, has a well-defined finite complexity in the generic case. Away from the cut locus, critical points of the restricted distance function are exactly points where the unique length-minimizing geodesic to $L$ is normal to $X$; Sard's theorem applied to the normal exponential map makes this set discrete, and o-minimal definability upgrades discreteness to a uniform bound $C_X$. For simple Schubert varieties the optimization is exactly solvable: the Eckart–Young construction produces critical points, the unique global minimizer spans the first $s$ principal vectors of the fixed plane $W$ together with the remaining principal vectors of $L$, and the global maximizers form a Grassmannian of dimension $(k-s)(n-k-s)$ that lies entirely in the cut locus.

Load-bearing premise

The uniform bound depends on the unproved assertion that the set of pairs $(S,L)$ with $S$ critical for $\delta_L|_X$ is definable in the paper's o-minimal structure; if that tameness fails, the generic finiteness of GDC could collapse.

Editorial extensions

If this is right

For any smooth subanalytic submanifold $X\subset G(k,n)$, the number of candidate nearest points that must be checked is bounded by a constant independent of the query plane $L$.
For smooth hypersurfaces of degree $d$, $\mathrm{GDC}(X)\le c_1(k,n)d^{c_2(k,n)}$, so the complexity grows at most polynomially in the degree with constants depending only on $k,n$.
For simple Schubert varieties $\Omega_s$, the Eckart–Young construction gives $\binom{k}{s}$ distinct critical points away from the cut locus, so $\mathrm{GDC}(\Omega_s)\ge\binom{k}{s}$.
Generic minimizers never lie on the cut locus, so the non-differentiable part of the distance function can be ignored when solving the nearest-point problem.
The global maximizers of distance to $\Omega_s$ form a Grassmannian of dimension $(k-s)(n-k-s)$, all lying on the cut locus, which explains why the cut locus must be excluded from the definition of GDC.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Our inference: the same subdifferential and o-minimal machinery should define an analogous complexity for partial flag manifolds, replacing principal angles by iterated flag angles.
Our inference: the existential bound in Theorem 42 and the huge Pfaffian constants suggest that the practical value of GDC lies in qualitative finiteness, not in the size of the bound; computing exact GDC for a concrete variety will likely need a geometric rather than Pfaffian route.
Our inference: for small examples such as $\Omega_s\subset G(3,6)$, numerical sampling of generic $L$ and counting critical points outside the cut locus could test whether the Eckart–Young points are the only critical points there, which the paper does not fully settle for global counts.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Short version: this paper introduces the Grassmann Distance Complexity (GDC), a sensible analogue of the Euclidean Distance Degree for subanalytic subsets of the Grassmannian, and it solves the nearest point problem for simple Schubert varieties with an explicit Eckart–Young type minimizer. The main theorems are genuinely new and mostly well proved. The soft spot is a specific unproved definability assertion in Theorem 42, which is load-bearing for the finiteness of GDC. It is likely fixable, but as written it is a gap.

What is actually new: the definition of GDC, the subdifferential analysis at the cut locus (the convex hull of O(j) description is nice), the proof that minima generically avoid the cut locus, the uniform finiteness theorem, and the nonlinear Eckart–Young theorem for Schubert varieties. The explicit unique minimizer, with distance (θ_1² + ... + θ_s²)^{1/2}, is a clean and surprising result. The family of global maximizers on a Grassmannian of dimension (k−s)(n−k−s) is also very concrete. The paper is honest: Remark 60 admits the authors do not yet understand why the Eckart–Young critical points are exactly the Grassmannian critical points, and the Pfaffian bounds are acknowledged to be astronomically large but existentially meaningful.

Where the soft spots are: the stress-test note is on target. In Theorem 42 the set A of critical pairs is asserted definable with no proof and no reference. The claim is plausible—away from the cut locus the inverse exponential map is definable, and derivatives of definable functions in o-minimal structures are definable—but this is not automatic and the paper does not supply the argument. Since the entire definition of GDC as a finite number depends on Theorem 42, this needs to be fixed. Similarly, the noncompact subanalytic case in Theorem 41 is dismissed with a one-line 'finiteness follows from o-minimality' that deserves a sentence or two more.

Everything else looks solid. The Schubert variety analysis is careful, the genericity arguments are standard, and the non-semialgebraicity example is convincing.

Who should read this: anyone working in metric algebraic geometry or on optimization over Grassmannians. It deserves a serious referee. The right request is: repair the definability gap (or cite a theorem that covers it), and the paper will be in good shape.

Referee Report

3 major / 5 minor

Summary. The paper introduces the Grassmann Distance Complexity (GDC) for subanalytic submanifolds of the real Grassmannian G(k,n), counting, for a generic external plane L, the number of critical points of the intrinsic distance function δ_L restricted to X, excluding points on the cut locus. The main theoretical results are: a description of the Clarke subdifferential of δ_L (Theorem 30); a proof that the number of critical points outside the cut locus is generically finite and uniformly bounded (Theorems 41 and 42); a Pfaffian bound on GDC for algebraic hypersurfaces (Theorem 49); and an analysis of the nearest-point problem for simple Schubert varieties, including a nonlinear Eckart-Young theorem identifying the unique global minimizer and an explicit family of global maximizers (Theorems 54, 61, 64).

Significance. If the gaps identified below are repaired, the paper makes a valuable contribution: it proposes a natural Riemannian analogue of Euclidean Distance Degree, correctly identifies the role of the cut locus and of Clarke critical points, and gives an explicit and apparently correct solution of the nearest-point problem for simple Schubert varieties. The subdifferential computation in Theorem 30 and the Eckart-Young critical points in Theorem 54 are concrete and useful, and the explicit global minimizer and maximizer in Theorems 61 and 64 are appealing. The Pfaffian bound in Theorem 49, though astronomically large, is a genuine effectivity statement. The work is not a repackaging of known results: the o-minimal framework, the treatment of the cut-locus strata, and the Schubert-variety analysis are new.

major comments (3)

[Section 3.5 (Theorem 42)] The proof of Theorem 42 asserts, immediately after the definition of A = {(S,L) in X×G(k,n) | S notin cut(L), S critical for δ_L|_X}, that 'By Proposition 23 and the definability of X, it follows that also the set A is definable.' This assertion is load-bearing: the o-minimal trivialization argument that produces the uniform bound C_X and the finiteness of GDC applies only if A is definable. Proposition 23 establishes that the principal angles and the distance function δ are globally subanalytic, but criticality of S for δ_L|_X involves the gradient of δ_L at S, which is the tangent vector obtained from the inverse exponential map exp_L^{-1}(S), and the orthogonality of that gradient to T_S X. The paper neither proves definability of this gradient map nor provides a reference for this step. The same missing definability is needed in Theorem 41 for noncompact subanalytic X, where the sentence 'finiteness follows from o-minimality' presupposes that the critical-point set is definable. Please supply a proof or a precise citation for the definability of the critical-pair set.
[Section 3.7 (Theorem 49)] The semi-Pfaffian set S is defined by the Lagrange multiplier equations (36) together with the maximal-rank condition rank([∂(U,V) p̃; D_(U,V)F]) = k(k+1). These equations are necessary only at regular points of the constraint Y = ~exp_L^{-1}(X\cut(L)). The parametrization Φ(U,V,μ) = U diag(μ) V^T is not a submersion at points with repeated singular values, so Y can be singular there, and a critical point of δ_L|_X whose every SVD lift lies in such a singular locus would not be represented in S. The proof does not show that every critical point admits a lift satisfying the maximal-rank condition (37), nor does it treat the repeated-singular-value case separately. Since the estimate GDC(X) ≤ b0(S) in (40) requires every critical point to have at least one preimage in S, this gap affects the validity of Theorem 49 as stated.
[Definition 43 / Theorem 42] The Grassmann Distance Complexity is defined as a maximum over 'generic' L, but the proof of Theorem 42 only produces a definable set ~M, the union of strata with zero-dimensional fibers, on which the count is bounded by C_X; the complement of ~M has codimension at least one. It is not stated that the count is constant on each connected component of ~M, nor is the meaning of 'generic' made precise, for example as belonging to a nonempty Zariski-open or definable dense subset. Without these clarifications, GDC(X) is not unambiguously a single number. The trivialization theorem does imply constancy of the fiber cardinality on each stratum, so this should be stated explicitly in the definition or in the proof of Theorem 42.

minor comments (5)

[Throughout] There are numerous typos, including 'Grassmann Dist ance' in the title, 'isomoprhism' in Definitions 13 and 14, 'Riemmanian sphere' in Section 3.6, and 'π 4 2' in the statement of Theorem 64, which should be π^2/4.
[Theorem 30] In equation (22), the symbol W is used both for the orthogonal matrix variable in the convex hull and later for the fixed subspace W defining Schubert varieties; this notational clash should be resolved.
[Section 3.7 (Theorem 49)] The sentence 'every connected component of S corresponds to exactly one critical point' is imprecise; the intended statement is that the continuous map from S to X sends each connected component to a single critical point and that every critical point has at least one preimage, so that GDC(X) ≤ b0(S).
[Theorem 61] In the uniqueness proof, the assertion that a minimizer E is an orthogonal direct sum E = E_L ⊕ E_W is not justified in the text. It follows from the equality conditions in the interlacing argument together with the uniqueness of principal vectors for distinct angles, but this step should be spelled out.
[Introduction and Theorem 5] The paper assumes k ≤ n−k from Section 2.2 onward, but Theorem 5 and some introductory statements claim results 'for every 0≤k≤n'; the two statements should be harmonized.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the central GDC theorems are derived from independent definitions and external tools, with self-citations only as background and one unproved definability gap that is a correctness risk rather than a circular reduction.

full rationale

The paper's central claims are not circular: the Grassmann Distance Complexity is defined as a generic maximum of critical-point counts, and Theorem 42 proves uniform finiteness via o-minimal trivialization rather than assuming it. The Eckart-Young transfer (Theorems 54 and 61) is proved by explicit geodesic and principal-angle computations, not by fitting parameters or renaming the desired conclusion. Self-citations to [18] (Grassmannian metric facts) and [11] (a standard subdifferential lemma) are background results external to the paper's target statements, and they are not used to import the paper's conclusions. The only significant gap is non-circular: in Theorem 42 the set A of critical pairs is asserted to be definable ('By Proposition 23 and the definability of X, it follows that also the set A is definable') without proving that the relevant gradient/orthogonality criticality condition is definable, and Theorem 41 says 'finiteness follows from o-minimality' for noncompact subanalytic X without a full argument. These are missing-support/correctness risks, not reductions by construction, so they do not raise the circularity score.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The central result introduces no new physical entities and fits no parameters. It depends on standard machinery (Clarke subdifferential, o-minimality, Pfaffian bounds) plus one unproved definability assertion.

assumptions (7)

domain assumption The Grassmannian G(k,n) is equipped with the unique (up to scale) orthogonally invariant Riemannian metric normalized so that the quotient map O(n) → G(k,n) is a Riemannian submersion.
This metric defines the distance function under study; it is standard and cited to [18].
domain assumption The principal angles and the Grassmann distance are globally subanalytic (definable in the o-minimal structure of globally subanalytic sets).
Used to apply o-minimal uniform finiteness. Proved in Proposition 23.
standard math Clarke's subdifferential theory applies to locally Lipschitz functions and the critical point condition 0 ∈ ∂f is used.
Standard nonsmooth analysis, cited to [3].
standard math Sard's Lemma is used to show generic L is a regular value of the normal exponential map.
Used in Lemma 40.
standard math The convex hull of the orthogonal group O(j) is the set of matrices with operator norm ≤ 1.
Used in Theorem 30 to describe the subdifferential; cited to [14,16].
ad hoc to paper The set of pairs (S,L) with S critical for δ_L|_X is definable in an o-minimal structure.
This assertion is made without proof in the proof of Theorem 42; it is load-bearing for the uniform bound. It is not obviously standard and no reference is given.
domain assumption We assume k ≤ n−k without loss of generality.
Since G(k,n) is isomorphic to G(n−k,n), this simplifies notation. Stated in Section 2.2.

how reviews work

0 comments

Cite this review

Pith. "Pith review of The Grassmann distance complexity." pith.science (2026). https://pith.science/paper/E3TUB4K4

@misc{pith2026241116589,
  author       = {Pith},
  title        = {Pith review of: The Grassmann distance complexity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/E3TUB4K4}},
  note         = {Machine review of arXiv:2411.16589}
}

abstract

Motivated by the concept of Euclidean Distance Degree, which measures the complexity of finding the nearest point to an algebraic set in Euclidean space, we introduce the notion of Grassmann Distance Complexity (GDC). This concept quantifies the complexity of solving the nearest point problem for subanalytic sets in the Grassmannian, using the intrinsic Riemannian distance. Unlike the Euclidean case, the Grassmannian distance is neither smooth nor semialgebraic, and its study requires using Lipschitz critical point theory and o-minimal geometry. We establish fundamental properties of GDC, including computable bounds for real algebraic varieties and conditions ensuring the finiteness of critical points. Our results also include a nonlinear version of the classical Eckart-Young theorem, which characterizes critical points of the distance function from a generic $k$-plane to simple Schubert varieties.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

22 extracted references · 20 canonical work pages

[1]

Thomas Bendokat, Ralf Zimmermann, and P.-A. Absil. A Gra ssmann manifold handbook: basic geometry and computational aspects. Adv. Comput. Math. , 50(1):Paper No. 6, 51, 2024

work page 2024
[2]

˙Ake Bj¨ orck and Gene H. Golub. Numerical methods for computi ng angles between linear subspaces. Math. Comp. , 27:579–594, 1973

work page 1973
[3]

F. H. Clarke. Optimization and nonsmooth analysis , volume 5 of Classics in Applied Mathematics . Society for Industrial and Applied Mathematics (SIAM), Phi ladelphia, PA, second edition, 1990

work page 1990
[4]

Riemannian geometry

Manfredo Perdig˜ ao do Carmo. Riemannian geometry . Mathematics: Theory & Applications. Birkh¨ auser Boston, Inc., Boston, MA, 1992. Translated from the second Portuguese edition by Francis Flaherty

work page 1992
[5]

Jan Draisma, Emil Horobet ¸, Giorgio Ottaviani, Bernd St urmfels, and Rekha R. Thomas. The Eu- clidean distance degree of an algebraic variety. Found. Comput. Math. , 16(1):99–149, 2016

work page 2016
[6]

The approximation of one matr ix by another of lower rank

Carl Eckart and Gale Young. The approximation of one matr ix by another of lower rank. Psychome- trika, 1(3):211–218, 1936

work page 1936
[7]

Feh´ er and´Akos K

L´ aszl´ o M. Feh´ er and´Akos K. Matszangosz. Real solutions of a problem in enumerat ive geometry. Period. Math. Hungar. , 73(2):137–156, 2016

work page 2016
[8]

Young tableaux, volume 35 of London Mathematical Society Student Texts

William Fulton. Young tableaux, volume 35 of London Mathematical Society Student Texts . Cambridge University Press, Cambridge, 1997. With applications to re presentation theory and geometry. THE GRASSMANN DISTANCE COMPLEXITY 41

work page 1997

Show all 22 references

[9]

Complexity of co mputations with Pfaﬃan and Noetherian functions

Andrei Gabrielov and Nicolai Vorobjov. Complexity of co mputations with Pfaﬃan and Noetherian functions. In Normal forms, bifurcations and ﬁniteness problems in diﬀere ntial equations, volume 137 of NATO Sci. Ser. II Math. Phys. Chem. , pages 211–250. Kluwer Acad. Publ., Dordr...

2004
[10]

I. M. Gelfand, M. M. Kapranov, and A. V. Zelevinsky. Discriminants, resultants and multidimensional determinants. Modern Birkh¨ auser Classics. Birkh¨ auser Boston, Inc., Boston, MA, 2008. Reprint of the 1994 edition

2008
[11]

Morse theory of euclidean dis- tance functions and applications to real algebraic geometr y

Andrea Guidolin, Antonio Lerario, Isaac Ren, and Marti na Scolamiero. Morse theory of euclidean dis- tance functions and applications to real algebraic geometr y. https://arxiv.org/abs/2402.08639, 2024

2024 arXiv
[12]

Morris W. Hirsch. Diﬀerential topology, volume 33 of Graduate Texts in Mathematics . Springer-Verlag, New York, 1994. Corrected reprint of the 1976 original

1994
[13]

Horn and Charles R

Roger A. Horn and Charles R. Johnson. Topics in matrix analysis . Cambridge University Press, Cam- bridge, 1991

1991
[14]

Generalized power method for sparse principal component analysis

Michel Journ´ ee, Yurii Nesterov, Peter Richt´ arik, and Rodolphe Sepulchre. Generalized power method for sparse principal component analysis. J. Mach. Learn. Res. , 11:517–553, 2010

2010
[15]

S. E. Kozlov. Geometry of real Grassmannian manifolds. I, II, III. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) , 246:84–107, 108–129, 197–198, 1997

1997
[16]

Absil Kyle A

P.-A. Absil Kyle A. Gallivan. Note on the convex hull of t he stiefel manifold. preprint on webpage at https://www.math.fsu.edu/%7Ealuffi/archive/paper386.pdf
[17]

John M. Lee. Introduction to smooth manifolds , volume 218 of Graduate Texts in Mathematics . Springer, New York, second edition, 2013

2013
[18]

A. Lerario. Lectures on metric algebraic geometry. pre print on webpage at https://drive.google.com/file/d/1yTaym4-NUwdjHymy0FUewYuzSG_Tl8FA/view
[19]

Tame topology and o-minimal structures , volume 248 of London Mathematical Society Lecture Note Series

Lou van den Dries. Tame topology and o-minimal structures , volume 248 of London Mathematical Society Lecture Note Series . Cambridge University Press, Cambridge, 1998

1998
[20]

Diﬀerential geometry of Grassmann man ifolds

Yung-chow Wong. Diﬀerential geometry of Grassmann man ifolds. Proc. Nat. Acad. Sci. U.S.A., 57:589– 594, 1967

1967
[21]

Schubert varieties and distance s between subspaces of diﬀerent dimensions

Ke Ye and Lek-Heng Lim. Schubert varieties and distance s between subspaces of diﬀerent dimensions. SIAM J. Matrix Anal. Appl. , 37(3):1176–1197, 2016

2016
[22]

Betti numbers of semi-Pfaﬃan sets

Thierry Zell. Betti numbers of semi-Pfaﬃan sets. J. Pure Appl. Algebra , 139(1):323–338, 1999. (Lerario) Scuola Internazionale Superiore di Studi A vanzati (SISSA) , via Bonomea, 265, 34136 Trieste, Italy Email address : lerario@sissa.it (Rosana) Scuola Internazionale Superior...

1999

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [1]

Thomas Bendokat, Ralf Zimmermann, and P.-A. Absil. A Gra ssmann manifold handbook: basic geometry and computational aspects. Adv. Comput. Math. , 50(1):Paper No. 6, 51, 2024

work page 2024

[2] [2]

˙Ake Bj¨ orck and Gene H. Golub. Numerical methods for computi ng angles between linear subspaces. Math. Comp. , 27:579–594, 1973

work page 1973

[3] [3]

F. H. Clarke. Optimization and nonsmooth analysis , volume 5 of Classics in Applied Mathematics . Society for Industrial and Applied Mathematics (SIAM), Phi ladelphia, PA, second edition, 1990

work page 1990

[4] [4]

Riemannian geometry

Manfredo Perdig˜ ao do Carmo. Riemannian geometry . Mathematics: Theory & Applications. Birkh¨ auser Boston, Inc., Boston, MA, 1992. Translated from the second Portuguese edition by Francis Flaherty

work page 1992

[5] [5]

Jan Draisma, Emil Horobet ¸, Giorgio Ottaviani, Bernd St urmfels, and Rekha R. Thomas. The Eu- clidean distance degree of an algebraic variety. Found. Comput. Math. , 16(1):99–149, 2016

work page 2016

[6] [6]

The approximation of one matr ix by another of lower rank

Carl Eckart and Gale Young. The approximation of one matr ix by another of lower rank. Psychome- trika, 1(3):211–218, 1936

work page 1936

[7] [7]

Feh´ er and´Akos K

L´ aszl´ o M. Feh´ er and´Akos K. Matszangosz. Real solutions of a problem in enumerat ive geometry. Period. Math. Hungar. , 73(2):137–156, 2016

work page 2016

[8] [8]

Young tableaux, volume 35 of London Mathematical Society Student Texts

William Fulton. Young tableaux, volume 35 of London Mathematical Society Student Texts . Cambridge University Press, Cambridge, 1997. With applications to re presentation theory and geometry. THE GRASSMANN DISTANCE COMPLEXITY 41

work page 1997