---
title: Shady Norm in Finite-Dimensional Spaces
url: https://www.emergentmind.com/topics/shady-norm
type: topic
---

# Shady Norm in Finite-Dimensional Spaces

Searching arXiv for recent papers relevant to “Shady Norm,” especially the finite-dimensional normed-space notion of shadiness and related nonstandard norms.
A **shady norm** is a norm on a finite-dimensional real vector space for which norm-one projections of intermediate rank fail to exist. In the formulation developed for \(X=(\mathbb{R}^d,\nu)\), the obstruction is quantified by the **shadiness constant** \(s_k(\nu)\), defined as the infimum of the operator norms of all rank-\(k\) projections, and by the global constant \(s(\nu)=\min_{k=2,\ldots,d-1}s_k(\nu)\). A norm is called **shady in dimension \(k\)**, or simply **shady**, when \(s_k(\nu)>1\). This terminology arises from the contrast with rank-one projections, which always admit norm one by the Hahn–Banach theorem, whereas projections of higher rank may be uniformly bounded away from one [2509.14819]. In a broader combinatorial literature, “shady” also describes nonstandard norms on sets whose behavior departs sharply from classical size-based norms, especially through failures of triangle inequalities or highly nonlocal dependence on the underlying combinatorial structure [1811.02137].

## 1. Concept and formal definition

Let \(X=(\mathbb{R}^d,\nu)\) be a finite-dimensional normed space with unit ball \(B_\nu\). For a projection \(P:\mathbb{R}^d\to\mathbb{R}^d\) of rank \(k\), the operator norm with respect to \(\nu\) is
\[
\nu(P)=\sup_{x\ne 0}\frac{\nu(Px)}{\nu(x)}.
\]
The **shadiness constant in dimension \(k\)** is
\[
s_k(\nu):=\inf\{\,\nu(P):P\text{ is a rank-}k\text{ projection}\,\},
\]
and the **global shadiness constant** is
\[
s(\nu):=\min_{k=2,\ldots,d-1}s_k(\nu).
\]
The extremal constants over all norms on \(\mathbb{R}^d\) are
\[
s_k(d)=\sup_\nu s_k(\nu),\qquad s(d)=\sup_\nu s(\nu).
\]
A norm is **shady in dimension \(k\)** if \(s_k(\nu)>1\) [2509.14819].

The key asymmetry between rank one and higher rank is structural. By the Hahn–Banach theorem, every normed space admits rank-one projections with operator norm one. Bosznay and Garay showed that for every \(d\ge 3\) there exist \(d\)-dimensional normed spaces \(X\) for which all projections of rank \(k\), with \(2\le k\le d-1\), have norm larger than or equal to some constant \(c>1\) [2509.14819]. The shadiness constant is the maximal such uniform lower bound.

This notion concerns the geometry of projections rather than the absolute size of vectors. A plausible implication is that shadiness measures how far a normed space is from permitting norm-preserving linear compression onto nontrivial subspaces. In that sense, it is a projection-theoretic invariant of the unit ball.

## 2. Historical setting and projection-theoretic motivation

The modern formulation of shadiness is tied to work of Bosznay and Garay on norms and projections in finite-dimensional spaces. Their result establishes the existence of spaces in which every projection of intermediate rank has norm strictly larger than one [2509.14819]. The later optimization-based treatment turns this qualitative statement into a quantitative program: determine explicit lower bounds for \(s_k(\nu)\), construct concrete shady norms, and understand what geometric complexity a unit ball must possess in order to be shady.

In dimension three, the problem is especially sharp. The rank of interest is \(k=2\), since rank one is trivial by Hahn–Banach and rank three is the identity. The 2025 paper “Lower Bounds for the Shadiness Constant of Finite-Dimensional Normed Spaces” develops two central directions: first, explicit computer-assisted lower bounds for \(s_2(\nu)\); second, a combinatorial classification result excluding shadiness for centrally symmetric convex polytopes with too few vertices [2509.14819].

The same word “shady” appears in a separate combinatorial tradition on norms on sets. There, the term does not denote projection constants, but rather captures behavior that is “not norm-like” in the classical analytic sense: norms may decrease when sets enlarge, triangle inequalities may fail, and values may depend on structural constraints such as coloring, covering, or Hall-type selection properties [1811.02137]. This broader usage clarifies that “shady” is not a single standardized technical term across mathematics; rather, it marks families of norm-like quantities whose utility comes precisely from their atypical behavior.

## 3. Quantitative results in finite-dimensional normed spaces

The principal explicit result currently recorded in the provided literature is a three-dimensional construction with a certified lower bound. There exists a norm on \(\mathbb{R}^3\), whose unit ball is a centrally symmetric convex polytope with \(12\) vertices, such that every rank-\(2\) projection has operator norm at least \(1.01\); equivalently, the shadiness constant is at least \(1.01\) [2509.14819].

The unit ball \(I\) is the convex hull of the \(12\) points
\[
\begin{aligned}
\{ &(1,a,c),\ (1,b,c),\ (c,1,a),\ (c,1,b), \\
&(a,c,1),\ (b,c,1), \\
&-(1,a,c),\ -(1,b,c),\ -(c,1,a),\ -(c,1,b),\ -(a,c,1),\ -(b,c,1)\}
\end{aligned}
\]
where
\[
a=-\frac35,\qquad b=-\frac15,\qquad c=\frac1{10}.
\]
The paper states that the shape is centrally symmetric, almost but not quite an icosahedron, and carefully perturbed to maximize the minimal projection norm [2509.14819].

A second sharp result concerns minimal combinatorial complexity. Let \(C\subset\mathbb{R}^3\) be a centrally symmetric convex polytope with at most \(10\) vertices. Then there exists a rank-\(2\) projection \(P\) with \(\nu(P)=1\); in other words, \(C\) is not shady [2509.14819]. This confirms a conjecture of Bosznay and Garay asserting that shady centrally symmetric convex polytopes in \(\mathbb{R}^3\) require at least \(12\) vertices.

The known quantitative landscape reported in the source may be summarized as follows.

| Space/unit ball | Statement |
|---|---|
| Any norm in \(\mathbb{R}^3\) | best known upper bound \(s_2\le \frac43-0.0007\) |
| Existence result (Kobos 2023) | \(s_2\ge 1+9.32\times 10^{-22}\) |
| 12-vertex polytope | \(s_2\ge 1.01\) |
| Polytopes with \(\le 10\) vertices | not shady |

These results show that the existence problem and the explicit-construction problem are distinct. Existence of shady norms is comparatively easy, but nontrivial lower bounds and sharp polytope classifications require substantially finer methods [2509.14819].

## 4. Optimization formulations and proof certificates

For a centrally symmetric polytope \(C\subset\mathbb{R}^d\) with vertex set \(V\) and supporting hyperplane normals \(H\), the shadiness constant of rank \(k\) can be formulated as a polynomial optimization problem. Define
\[
M_{k,C}=\{\,(\alpha,P): P^2=P,\ \mathrm{tr}(P)=k,\ h^\top Pv\le \alpha\ \forall\, v\in V',\ h\in H\,\}.
\]
Then \(s_k(C)\) is obtained from the program
\[
\begin{array}{ll}
\text{Minimize:} & \alpha\\
\text{Subject to:} & (\alpha,P)\in M_{k,C}.
\end{array}
\]
This converts a geometric minimization over projections into a semialgebraic feasibility problem [2509.14819].

For rank \(d-1\), a projection may be parameterized by vectors \(u\) and \(w\), representing kernel and image normal, by
\[
Px=x-\frac{w^\top x}{w^\top u}u.
\]
The corresponding constraints become
\[
\begin{aligned}
&\text{Minimize: }\alpha\\
&\text{s.t. }w^\top u\ge 0,\quad u,w\in[-1,1]^d,\\
&\exists i:\ u_i\in\{-1,1\},\quad \exists j:\ w_j=1,\\
&h^\top v\, w^\top u-w^\top v\, h^\top u-\alpha\, w^\top u\le 0,\quad \forall\, v\in V',\ h\in H.
\end{aligned}
\]
In dimension three this specializes directly to the rank-\(2\) case [2509.14819].

The lower bound \(s_2(I)\ge 1.01\) was established by two independently checkable, computer-assisted proofs: one based on **sums of squares (SOS) certificates**, the other on **linear duality via Farkas’ Lemma**. In the SOS approach, infeasibility of a target level \(\alpha^*\) is certified through a weighted decomposition of the form
\[
-1=q_0+\sum_j q_j g_j+\sum_i p_i f_i,
\]
where the \(f_i\) are equality constraints and the \(g_j\) are inequalities defining the feasible set. The paper reports the use of SDP solvers for floating-point certificates, followed by rationalization and rounding for exact verification [2509.14819].

In the Farkas-based method, for fixed image normal \(w\), infeasibility is reduced to the existence of a nonnegative vector \(y\) satisfying \(A_{w,\alpha^*}^\top y=w\). Computations on a dense finite subset of the sphere then yield a uniform lower bound up to a small perturbation. The source states that code and proof certificates were made publicly available for independent verification [2509.14819].

These methods do not merely estimate the shadiness constant numerically; they deliver rigorous lower bounds. This suggests an interface between convex algebraic geometry, semialgebraic optimization, and Banach-space projection theory.

## 5. Geometry of shady polytopes

In the polytopal setting, shadiness is encoded in the interaction between facets, supporting normals, and the behavior of subspaces intersecting opposite facets. A simple sufficient condition reported in the source is the following: if all sets of \(d\) normals to facets of a centrally symmetric polytope \(C\) are linearly independent, and every \((d-1)\)-dimensional subspace intersects at least \(d\) pairs of opposite facets, then \(C\) is shady in dimension \(d-1\) [2509.14819].

The negative result for polytopes with at most \(10\) vertices is proved combinatorially and geometrically. The outline given in the source is that any centrally symmetric triangulated convex polytope in \(\mathbb{R}^3\) with \(10\) vertices on its boundary must contain a \(4\)-cycle of the form \((v,w,-v,-w)\). Supporting hyperplanes to facets containing the relevant edges then determine a two-dimensional subspace onto which there exists a projection of norm one. Hence shadiness fails [2509.14819].

The \(12\)-vertex example shows that the obstruction is not merely volumetric or symmetry-based; it is sensitive to the precise arrangement of vertices and facets. The source explicitly notes that the extremal polytope is “almost (but not quite) an icosahedron,” which indicates that near-regularity alone is insufficient to determine shadiness. A plausible implication is that local perturbations of polytope geometry can have global effects on the minimal norm of projections.

## 6. Related nonstandard norms on sets

A distinct but conceptually related line of work surveys several combinatorial norms on sets, emphasizing their deviation from standard analytic norms [1811.02137]. The paper “Properties of Norms on Sets” reviews four norms from Bartoszyński and Judah, Rosłanowski and Shelah, and Shelah. These norms are monotone, but some fail the triangle inequality or behave in ways that invert size intuition.

The surveyed examples are:

| Norm type | Key feature |
|---|---|
| Standard counting norm \(\mathrm{norm}_0(B)=|B|\) | conventional baseline |
| Exclusion norm \(\mathrm{norm}_1^{F,G}(A)=\frac{G-|A|+1}{F}\) | decreases as \(|A|\) increases |
| Subset norm \(\mathrm{norm}_2^{n,G}(\mathcal A)\) | minimal size of a set not contained in any member of \(\mathcal A\) |
| Graph coloring norm \(\mathrm{norm}_3(A)\) | tied to chromatic and splitting properties |
| Hall norm \(\mathrm{norm}_4(A)=H_N(\Delta_N(A))\) | selector/Hall-like behavior |

The **exclusion norm** is monotone but larger sets have smaller norm, and the inequality
\[
\mathrm{norm}_1(A\cup B)\le \mathrm{norm}_1(A)+\mathrm{norm}_1(B)
\]
can fail “spectacularly” [1811.02137]. The **subset norm** satisfies a triangle inequality and measures the minimal size of a subset of \(G\) not contained in any element of a family \(\mathcal A\). The **graph coloring norm** is defined recursively and is equivalent to a coloring condition; for families of edges, it equals the chromatic number. The source remarks that adding elements to \(A\) can sometimes sharply decrease the norm. The **Hall norm** is monotone, but triangle inequality fails: there are sets \(A,B\) with \(\mathrm{norm}_4(A)=\mathrm{norm}_4(B)=2\) while \(\mathrm{norm}_4(A\cup B)=N+1\), the maximum possible [1811.02137].

These combinatorial norms are “shady” in a different sense from shadiness constants of finite-dimensional normed spaces. Here the adjective refers to the fact that the norm-like quantity is not a size measure and may exhibit nonlocal, context-sensitive jumps. The source explicitly states that these “apparently ‘shady’ properties” are features rather than bugs for forcing applications [1811.02137]. This broader usage is relevant because it clarifies a common misconception: a “shady norm” need not always denote the projection-theoretic notion \(s_k(\nu)>1\); in some literature it denotes unusual combinatorial norms whose formal properties depart from classical norm axioms.

## 7. Interpretation, misconceptions, and current scope

Two points are fundamental.

First, **shadiness is not about failure of the norm axioms** in the finite-dimensional projection-theoretic setting. A shady norm on \(\mathbb{R}^d\) is an ordinary norm \(\nu\); what is exceptional is the projection geometry induced by \(\nu\), specifically that every rank-\(k\) projection for some \(2\le k\le d-1\) has operator norm strictly larger than one [2509.14819].

Second, **the term is context-dependent across the literature**. In finite-dimensional Banach-space geometry, shadiness is encoded by the constants \(s_k(\nu)\) and \(s(\nu)\). In combinatorial set-theoretic work, “shady” describes norm-like invariants with unusual monotonicity, additivity, covering, or coloring behavior [1811.02137]. Conflating these usages obscures rather than clarifies the underlying mathematics.

The current explicit state of the art in the supplied sources is strongest in dimension three. There is a concrete centrally symmetric \(12\)-vertex polytope with \(s_2\ge 1.01\), and there is a sharp obstruction showing that centrally symmetric convex polytopes with \(10\) or fewer vertices are not shady [2509.14819]. This suggests that the geometry of low-dimensional unit balls, especially their combinatorial complexity, plays a decisive role in achieving quantitatively nontrivial shadiness.

A plausible implication is that future advances will continue to combine geometric combinatorics with certified optimization, especially SOS and SDP-based methods, in order to sharpen lower bounds for \(s_k(d)\), construct new explicit examples, and understand the relationship between polytope structure and projection constants. Within the supplied literature, that synthesis is the defining methodological feature of the modern theory of shady norms in finite-dimensional spaces [2509.14819].

Source: https://www.emergentmind.com/topics/shady-norm