Shadiness Constant in Normed Spaces

• Shadiness constant is defined as the minimal operator norm of rank-k projections, characterizing how far these projections deviate from being contractive.
• Optimization techniques including SOS and LP duality yield provable lower bounds, exemplified by a 3D normed space where every rank-2 projection has norm at least 1.01.
• A combinatorial-geometric result shows that centrally symmetric 3D polytopes with 10 or fewer vertices cannot be shady, establishing a sharp vertex threshold.

The shadiness constant of a finite-dimensional normed space quantifies how far higher-rank projections must be from being contractive. For a normed space $X=(\mathbb{R}^d,\nu)$, it is defined from the minimal operator norms of rank-$k$ projections, in contrast to the rank-one case, where the Hahn–Banach theorem guarantees norm-one projections. Bosznay and Garay showed that for every $d\geq 3$ there exist $d$-dimensional normed spaces in which every projection of rank $k$, for $2\leq k\leq d-1$, has norm at least $c>1$; the maximal such constant is the shadiness constant. Recent work develops optimization-based methods that yield provable lower bounds, including an explicit $3$-dimensional example with shadiness constant at least $1.01$, and a nonexistence theorem for centrally symmetric $3$-dimensional polytopes with $k$0 or fewer vertices (Epperlein, 18 Sep 2025).

1. Definition and formal framework

Let $k$1 be a finite-dimensional real normed space, equivalently with centrally symmetric convex unit ball $k$2. For a linear operator $k$3, the operator norm with respect to $k$4 is

$k$5

For a polytopal norm with unit ball $k$6, the paper also writes

$k$7

The rank-$k$8 shadiness constant is

$k$9

and the global shadiness constant is

$d\geq 3$0

The associated extremal quantities are

$d\geq 3$1

A norm $d\geq 3$2 is called $d\geq 3$3-shady in dimension $d\geq 3$4 if $d\geq 3$5, and globally $d\geq 3$6-shady if this holds for all $d\geq 3$7 (Epperlein, 18 Sep 2025).

The terminology isolates a distinctly higher-rank phenomenon. By the Hahn–Banach theorem, every normed space admits rank-one projections with operator norm one. The shadiness constant measures the obstruction to extending that norm-one behavior to higher-rank projections. In the language of the paper’s summary, higher-rank projections in a shady space are necessarily “expansive,” in the sense that their operator norm is strictly greater than one (Epperlein, 18 Sep 2025).

2. Structural viewpoint and extremal questions

The central existence statement predates the recent quantitative advances: for every $d\geq 3$8 there are $d\geq 3$9-dimensional normed spaces $d$0 such that all projections of rank $d$1, with $d$2, have norm at least some constant $d$3. Although constructing such spaces is not difficult, few explicit estimates of their shadiness constants were available before the optimization-based approach developed in the paper (Epperlein, 18 Sep 2025).

Two structural properties organize the theory. First, the shadiness constant is invariant under invertible linear maps, so affinely equivalent unit balls define the same value. Second, the function $d$4 is continuous in the Banach–Mazur topology. These facts place the problem in the standard geometric framework of finite-dimensional Banach space theory and justify passing between norm descriptions and centrally symmetric convex bodies (Epperlein, 18 Sep 2025).

The extremal problem is therefore twofold. At the level of a fixed norm, one seeks lower bounds on $d$5 or $d$6. At the level of the dimension alone, one seeks lower and upper bounds on $d$7 and $d$8. The paper under discussion focuses on the first direction, with a particular emphasis on explicit three-dimensional polytopal norms and on rigorous certification of lower bounds rather than heuristic or purely numerical evidence (Epperlein, 18 Sep 2025).

3. Optimization formulations and proof certificates

For a polytopal norm with unit ball $d$9, the shadiness constant can be expressed as a constrained optimization problem. In the paper’s primal formulation, one minimizes $k$0 subject to the conditions that $k$1 is a projection of rank $k$2,

$k$3

together with facet-vertex inequalities

$k$4

for all facet normals $k$5 and vertices $k$6 of $k$7. In this form,

$k$8

For rank-$k$9 projections, the problem can be reduced to optimizing over pairs of vectors describing kernel and image, which simplifies the search space (Epperlein, 18 Sep 2025).

The paper develops two complementary certification strategies. The first uses sum-of-squares (SOS) techniques from real algebraic geometry. The feasibility region is encoded as a basic semialgebraic set, and an explicit SOS decomposition of $2\leq k\leq d-1$0 as a polynomial combination of the constraints serves as a certificate that the region is empty for a chosen $2\leq k\leq d-1$1. In particular, this yields rigorous lower bounds by proving infeasibility of all projections with norm below the target threshold (Epperlein, 18 Sep 2025).

The second strategy uses linear programming duality and a Farkas lemma perspective. For fixed images of projections, the method constructs dual certificates lying in cones generated by constraint vectors. The argument is then extended over all possible images by means of dense coverings of the sphere. The summary emphasizes the use of Carathéodory’s theorem for sparsity and rational arithmetic for exactness. The computational side is implemented in Julia with SumOfSquares.jl, Clarabel.jl, and HiGHs.jl, producing independently verifiable, machine-checkable certificates (Epperlein, 18 Sep 2025).

These methods are significant because the relevant optimization problems are nontrivial even in low dimension. The contribution is not only an existence proof, but a framework for obtaining provable numerical lower bounds.

4. Explicit three-dimensional construction

A central achievement is an explicit norm on $2\leq k\leq d-1$2 whose unit ball is a centrally symmetric convex polytope with $2\leq k\leq d-1$3 vertices,

$2\leq k\leq d-1$4

with

$2\leq k\leq d-1$5

For this norm, every rank-$2\leq k\leq d-1$6 projection has operator norm at least $2\leq k\leq d-1$7, i.e.

$2\leq k\leq d-1$8

Equivalently, the polytope defines a $2\leq k\leq d-1$9-dimensional normed space that is $c>1$0-shady in rank $c>1$1 (Epperlein, 18 Sep 2025).

The result is quantitative as well as constructive. The summary explicitly notes that this is an improvement over prior examples, where earlier lower bounds for $c>1$2 were extremely close to $c>1$3, for example $c>1$4 in work of Kobos from 2023. In that sense, the new example is the first explicit three-dimensional construction in the supplied literature with a visibly nontrivial certified gap above $c>1$5 (Epperlein, 18 Sep 2025).

The unit ball is polytopal rather than smooth, which is important methodologically. The facet-vertex optimization model, the SOS certificates, and the exact arithmetic aspects of the dual method are especially well adapted to polytopes. A plausible implication is that polytopal models provide a practical interface between finite-dimensional Banach space geometry and computer-assisted proof techniques.

5. Minimal-vertex obstruction in dimension three

The second principal theorem is a nonexistence result. Let $c>1$6 be a centrally symmetric convex polytope with at most $c>1$7 vertices. Then there always exists a rank-$c>1$8 projection $c>1$9 with

$3$0

Hence no such polytope, and no corresponding norm, is shady. The paper states that this confirms a conjecture of Bosznay and Garay from 1986 (Epperlein, 18 Sep 2025).

The proof uses a combinatorial-geometric argument. Any centrally symmetric triangulated convex polyhedron in $3$1 with $3$2 vertices must contain a $3$3-cycle formed by two pairs of antipodal vertices. The intersection of the unit ball with the plane spanned by such a pair is a parallelogram, and that parallelogram admits a projection of norm $3$4. This forces the ambient norm to fail shadiness in rank $3$5 (Epperlein, 18 Sep 2025).

Unit ball in $3$6	Projection conclusion	Shadiness consequence
Centrally symmetric polytope with at most $3$7 vertices	There exists a rank-$3$8 projection $3$9 with $1.01$0	Not shady
Explicit centrally symmetric polytope with $1.01$1 vertices	Every rank-$1.01$2 projection has norm at least $1.01$3	$1.01$4

This leaves a sharp gap in the supplied results: $1.01$5 or fewer vertices are impossible, while $1.01$6 vertices suffice. No statement about $1.01$7 vertices appears in the provided material.

6. Terminological scope and neighboring notions

Within the supplied literature, the phrase “shadiness constant” is context-dependent. In the normed-space setting treated here, it denotes the extremal lower bound on operator norms of higher-rank projections (Epperlein, 18 Sep 2025). In the summary of “Moser’s Shadow Problem,” the same label is attached to the bounded shadow function $1.01$8, which measures the minimum guaranteed number of vertices in a shadow of an $1.01$9-vertex convex polyhedron (Lagarias et al., 2013). In the summary of “Volume bounds for shadow covering,” the label is also used for the universal constant $3$0 in the volume inequality

$3$1

under a projection-covering hypothesis (Chen et al., 2011).

This indicates that the normed-space invariant $3$2 should be kept separate from combinatorial shadow-counting and from volume-ratio constants in convex geometry. The common lexical root is “shadow,” but the mathematical objects are different: operator norms of projections in finite-dimensional Banach spaces, vertex counts of projected polyhedra, and volume bounds under shadow covering. For technical work, the defining formulas are the safest disambiguation.

In the normed-space literature summarized here, the central open-ended direction is quantitative rather than terminological: explicit lower bounds remain scarce, and the recent contribution is notable precisely because it combines a concrete $3$3-vertex construction, a certified lower bound $3$4, and a sharp nonexistence theorem for $3$5-vertex centrally symmetric polytopes in $3$6 (Epperlein, 18 Sep 2025).