1 Loops

1.1 Introduction

In this chapter, we explain how to construct families of loops to feed into the corrugation process explained at the end of the introduction. A loop is a map defined on the circle $𝕊^1 = ℝ/ℤ$ with values in a finite-dimensional vector space. It can also freely be seen as $1$-periodic maps defined on $ℝ$.

Definition 1.1 ✓

The average of a loop $γ$ is $\barγ := \int _{𝕊^1} γ(s)\, ds$.

Throughout this document, $E$ and $F$ will denote finite-dimensional real vector spaces.

Definition 1.2 ✓

The support of a family $γ$ of loops in $F$ parametrized by $E$ is the closure of the set of $x$ in $E$ such that $γ_x$ is a constant loop.

All of this chapter is devoted to proving the following proposition.

Proposition 1.3

Let $U$ be an open set in $E$ and $K ⊆ U$ a compact subset and $U'$ an open neighborhood of $\overline U$. Let $Ω$ be a set in $E × F$ such that, for each $x$ in $U'$, $Ω_x := Ω ∩ (\{ x\} × F)$ is open and connected in $\{ x\} × F$.

Let $β$ and $g$ be maps from $E$ to $F$ that are smooth on $U'$. Assume that $β(x) ∈ Ω_x$ for all $x$ in $U'$, and $g(x) = β(x)$ near $K$.

If, for every $x$ in $U'$, $g(x)$ is in the convex hull of $Ω_x$, then there exists a smooth family of loops

\[ γ \! :E × [0, 1] × 𝕊^1 → F, (x, t, s) ↦ γ^t_x(s) \]

such that, for all $x$ in $U$, and all $(t, s) ∈ [0, 1] × 𝕊^1$

$γ^t_x(s) ∈ Ω_x$
$γ^0_x(s) = β(x)$
$\barγ^1_x = g(x)$
$γ^t_x(s) = β(x)$ if $x$ is near $K$.

Let us briefly sketch the geometric idea behind the above proposition if we pretend there is only one point $x$, and drop it from the notation, and also focus only on $γ^1$. By assumption, there is a finite collection of points $p_i$ in $Ω$ and $λ_i ∈ [0, 1]$ such that $g$ is the barycenter $\sum λ_i p_i$. Since $Ω$ is open and connected, there is a smooth loop $γ_0$ which goes through each $p_i$. The claim is that $g$ is the average value of $γ = γ_0 ∘ h$ for some self-diffeomorphism $h$ of $𝕊^1$. The idea is to choose $h$ such that $γ$ rushes to $p_1$, stays there during a time roughly $λ_1$, rushes to $p_2$, etc. But, in order to achieve average exactly $g$, it seems like $h$ needs to be a discontinuous piecewise constant map. The assumption that $g$ is in the interior of the convex hull gives enough slack to get away with a smooth $h$. Actually the conclusion would be false without this interior assumption.

In the previous proof sketch, there is a lot of freedom in constructing $γ$, which is problematic when trying to do it consistently when $x$ varies.

1.2 Preliminaries

In this section, $E$ is a real vector space with (finite) dimension $d$. We’ll need the Carathéodory lemma:

Lemma 1.4 Carathéodory’s lemma ✓

If a point $x$ of $E$ lies in the convex hull of a set $P$, then $x$ belongs to the convex hull of a finite set of affinely independent points of $P$.

Proof ▼

By assumption, there is a finite set of points $t_i$ in $P$ and weights $f_i$ such that $x = \sum f_i t_i$, each $f_i$ is non-negative and $\sum f_i = 1$. Choose such a set of points of minimum cardinality. We argue by contradiction that such a set must be affinely independent.

Thus suppose that there is some vanishing combination $\sum g_i t_i$ with $\sum g_i = 0$ and not all $g_i$ vanish. Let $S = \{ i | g_i {\gt} 0\} $. Let $i_0$ in $S$ be an index minimizing $f_i/g_i$. We shall obtain our contradiction by showing that $x$ belongs to the convex hull of the set $\{ t_i| i \ne i_0\} $, which has cardinality strictly smaller than $\{ t_i\} $.

We thus define new weights $k_i = f_i - g_i f_{i_0}/g_{i_0}$. These weights sum to $\sum f_i - (\sum g_i)f_{i_0}/g_{i_0} = 1$ and $k_{i_0} = 0$. Each $k_i$ is non-negative, thanks to the choice of $i_0$ if $i$ is in $S$ or using that $f_i$, $-g_i$ and $f_{i_0}/g_{i_0}$ are all non-negative when $i$ is not in $S$. It remain to compute

\begin{align*} \sum _{i ≠ i_0} k_i t_i & = \sum _i k_i t_i \\ & = \sum _i (f_i - g_i f_{i_0}/g_{i_0}) t_i \\ & = \sum _i f_i t_i - \left(\sum _i g_i t_i\right)f_{i_0}/g_{i_0}) \\ & = x \end{align*}

where we use $k_{i_0} = 0$ in the first equality.

Definition 1.5 ✓

A point $x$ in $E$ is surrounded by points $p_0$, …, $p_d$ if those points are affinely independent and there exist weights $w_i ∈ (0, 1)$ such that $x = \sum _i w_i p_i$.

Note that, in the above definition, the number of points $p_i$ is fixed by the dimension $d$ of $E$, and that the weights $w_i$ are the barycentric coordinates of $x$ with respect to the affine basis $p_0, \ldots , p_d$.

Lemma 1.6 ✓

Given an affine basis $b$ of $E$, the interior of the convex hull of $b$ is the set of points with strictly positive barycentric coordinates.

Proof ▼

For each $i$, let:

\[ w_i \! :E \to ℝ \]

be the $i^{\rm th}$ barycentric coordinate with respect to the basis $b$. Since $E$ is finite-dimensional, each $w_i$ is a continuous open map. For such a map, the operation of taking interior commutes with preimage, and so we have:

\begin{align*} \operatorname{IntConv}(b) & = \operatorname{Int}\left(\bigcap _i w_i^{-1}([0, \infty ))\right)\\ & = \bigcap _i \operatorname{Int}(w_i^{-1}([0, \infty ))\\ & = \bigcap _i w_i^{-1}(\operatorname{Int}([0, \infty ))\\ & = \bigcap _i w_i^{-1}((0, \infty )) \end{align*}

as required.

Lemma 1.7 ✓

Given a point $c$ of $E$ and a real number $t$, let:

\[ h^c_t \! :E \to E \]

be the homothety which dilates about $c$ by a scale of $t$.

Suppose $c$ belongs to the interior of a convex subset $C$ of $E$ and $t {\gt} 1$, then

\[ C \subseteq \operatorname{Int}(h^c_t(C)) \]

Proof ▼

Since $h^c_t$ is a homeomorphism with inverse $h^c_{t^{-1}}$, taking $s = t^{-1}$, the required result is equivalent to showing:

\[ h^c_s(C) \subseteq \operatorname{Int}(C) \]

where $s \in (0, 1)$.

Let $x$ be a point of $C$, we must show there exists an open neighborhood $U$ of $h^c_s(x)$, contained in $C$. In fact we claim:

\[ U = h^x_{1-s}(\operatorname{Int}(C)) \]

is such a set. Indeed $U$ is open since $h^x_{1-s}$ is a homeomorphism and $U$ contains $h^c_s(x)$ since:

\[ h^c_s(x) = h^x_{1-s}(c) \in h^x_{1-s}(\operatorname{Int}(C)) \]

since $c$ belongs to $\operatorname{Int}(C)$. Finally:

\[ h^x_{1-s}(\operatorname{Int}(C)) \subseteq h^x_{1-s}(C) \subseteq C \]

where the second inclusion follows since $C$ is convex and contains $x$.

Lemma 1.8 ✓

If a point $x$ of $E$ lies in the convex hull of an open set $P$, then it is surrounded by some collection of points belonging to $P$.

Proof ▼

It follows from Lemma 1.6 that we need only show that $E$ has an affine basis $b$ of points belonging to $P$ such that $x$ lies in the interior of the convex hull of $b$.

Carathéodory’s lemma 1.4 provides affinely independent points $p_0, \dots , p_k$ in $P$ such that $x$ belongs to their convex hull. Since $P$ is open, we may extend $p_i$ to an affine basis

\[ \hat b = \{ p_0, \ldots , p_d\} , \]

where all points still belong to $P$. Note that $x$ belongs to the convex hull of $\hat b$.

Now let $c$ be a point in the interior of the convex hull of $\hat b$ (e.g., the centroid) and for each $\epsilon {\gt} 0$, consider the homothety

\[ h_{1+\epsilon } \! :E \to E, \]

which dilates about $c$ by a scale of $1 + \epsilon $.

Since $\hat b$ is finite and contained in $P$, and $P$ is open, there exists $\epsilon {\gt} 0$ such that

\[ h_{1+\epsilon } (\hat b) \subseteq P. \]

We claim the required basis is:

\[ b = h_{1+\epsilon } (\hat b) \]

for any such $\epsilon $. Indeed, applying Lemma 1.7 to $\operatorname{Conv}(\hat b)$ we see:

\begin{align*} x \in \operatorname{Conv}(\hat b) & \subseteq \operatorname{Int}(h_{1+\epsilon } (\operatorname{Conv}(\hat b)))\\ & = \operatorname{Int}(\operatorname{Conv}(h_{1+\epsilon } (\hat b))) \end{align*}

as required.

Lemma 1.9 ✓

For every $x$ in $E$ and every collection of points $p ∈ E^{d+1}$ surrounding $x$, there is a neighborhood $U$ of $\{ (x, p)\} $ and a function $w \! :E × E^{d+1} → ℝ^{d+1}$ such that, for every $(y, q)$ in $U$,

$w$ is smooth at $(y, q)$
$w(y, q) {\gt} 0$
$\sum _{i=0}^d w_i(y, q) = 1$
$y = \sum _{i=0}^d w_i(y, q)q_i$

Proof ▼

Let:

\begin{align*} A = E \times \{ q \in E^{d+1} ~ |~ \mbox{$q$ is an affine basis for $E$} \} , \end{align*}

and define:

\begin{align*} w \! :A & \to ℝ^{d+1}\\ (y, q) & \mapsto \mbox{barycentric coordinates of $y$ with respect to $q$}. \end{align*}

If we fix an affine basis $b$ of $E$, we may express $w$ as a ratio of determinants in terms of coordinates relative to $b$. More precisely, by Cramer’s rule, if $0 \le i \le d$ and $w_i$ is the $i^{\rm th}$ component of $w$, then:

\begin{align*} w_i (y, q) = \det M_i (y, q) / \det N (q) \end{align*}

where $N(q)$ is the $(d+1)\times (d+1)$ matrix whose columns are the barycentric coordinates of the components of $q$ relative to $b$, and $M_i (y, q)$ is $N(q)$ except with column $i$ replaced by the barycentric coordinates of $y$ relative to $b$.

Since determinants are smooth functions and $(y, q) \mapsto \det N(q)$ is non-vanishing on $A$, $w$ is smooth on $A$.

Finally define:

\begin{align*} U = w^{-1}((0, \infty )^{d+1}), \end{align*}

and note that $U$ is open in $A$, since it is the preimage of an open set under the continuous map $w$. In fact since $A$ is open, $U$ is open as a subset of $E \times E^{d+1}$. Note that $(x, p) \in U$ since $p$ surrounds $x$.

We may extend $w$ to $E \times E^{d+1}$ by giving it any values at all outside $A$.

1.3 Constructing loops

1.3.1 Surrounding families

It will be convenient to introduce some more vocabulary.

Definition 1.10 ✓

We say a loop $γ$ surrounds a vector $v$ if $v$ is surrounded by a collection of points belonging to the image of $γ$. Also, we fix a base point $0$ in $𝕊^1$ and say a loop is based at some point $b$ if $0$ is sent to $b$.

The first main task in proving Proposition 1.3 is to construct suitable families of loops $γ_x$ surrounding $g(x)$, by assembling local families of loops. Those will then be reparametrized to get the correct average in the next section. In this section, we will work only with continuous loops. This will make constructions easier and we will smooth those loops in the end, taking advantage of the fact that $Ω$ and the surrounding condition are open.

Thanks to Carathéodory’s lemma, constructing one such loop with values in some open $O$ is easy as soon as $v$ belongs to the convex hull of $O$.

Lemma 1.11 ✓

If a vector $v$ is in the convex hull of a connected open subset $O$ then, for every base point $b ∈ O$, there is a continuous family of loops $γ \! :[0, 1] × 𝕊^1 → E, (t, s) ↦ γ^t(s)$ such that, for all $t$ and $s$:

$γ^t$ is based at $b$
$γ^0(s) = b$
$γ^t(s) ∈ O$
$γ^1$ surrounds $v$

Proof ▼

Since $O$ is open, Lemma 1.8 gives points $p_i$ in $O$ surrounding $x$. Since $O$ is open and connected, it is path connected. Let $λ \! :[0, 1] → Ω_x$ be a continuous path starting at $b$ and going through the points $p_i$. We can concatenate $λ$ and its opposite to get $γ^1$, say $γ^1(s) = λ((1-\cos 2πs)/2)$. This is a round-trip loop: it back-tracks when it reaches $λ(1)$ at $s = 1/2$. We then define $γ^t$ as the round-trip that stops at $s = t/2$, stays still until $s = 1-t/2$ and then backtracks.

Definition 1.12 ✓

A continuous family of loops $γ \! :E × [0, 1] × 𝕊^1 → F, (x, t, s) ↦ γ^t_x(s)$ surrounds a map $g \! :E → F$ with base $β \! :E → F$ on $U ⊆ E$ in $Ω ⊆ E × F$ if, for every $x$ in $U$, every $t ∈ [0, 1]$ and every $s ∈ 𝕊^1$,

$γ^t_x$ is based at $β(x)$
$γ^0_x(s) = β(x)$
$γ^1_x$ surrounds $g(x)$
$(x,γ^t_x(s)) ∈ Ω$.

The space of such families will be denoted by $\operatorname{\mathcal{L}}(g, β, U, Ω)$.

Families of surrounding loops are easy to construct locally.

Lemma 1.13 ✓

Assume $Ω$ is open and connected over some neighborhood of $x_0$. If $g(x)$ is in the convex hull of $Ω_x$ for $x$ near $x_0$ then there is a continuous family of loops defined near $x_0$, based at $β$, taking value in $Ω$ and surrounding $g$.

Proof ▼

In this proof we don’t mention the $t$ parameter since it plays no role, but it is still there. Lemma 1.11 gives a loop $γ$ based at $β(x_0)$, taking values in $Ω_{x_0}$ and surrounding $g(x_0)$. We set $γ_x(s) = β(x) + (γ(s) - β(x_0))$. Each $γ_x$ takes values in $Ω_x$ because $Ω$ is open over some neighborhood of $x_0$. Lemma 1.9 guarantees that this loop surrounds $g(x)$ for $x$ close enough to $x_0$.

The difficulty in constructing global families of surrounding loops is that there are plenty of surrounding loops and we need to choose them consistently. The key feature of the above definition is that the $t$ parameter not only allows us to cut out the corrugation process in the next chapter, but also brings a “satisfied or refund” guarantee, as explained in the next lemma.

Lemma 1.14 ✓

Each $\operatorname{\mathcal{L}}(g, β, U, Ω)$ is path connected: for every $γ_0$ and $γ_1$ in $\operatorname{\mathcal{L}}(g, β, U, Ω)$, there is a continuous map $δ \! :[0, 1] × E × [0, 1] × 𝕊^1 → F, (τ, x, t, s) ↦ δ^t_{τ, x}(s)$ which interpolates between $γ_0$ and $γ_1$ in $\operatorname{\mathcal{L}}(g, β, U, Ω)$.

Proof ▼

Let $ρ$ be the piecewise affine map from $ℝ$ to $ℝ$ such that $ρ(τ) = 1$ if $τ ≤ 1/2$, $ρ$ is affine on $[1/2, 1]$, $ρ(τ) = 0$ if $τ ≥ 1$. We set

\[ δ_{τ, x}^t(s) = \begin{cases} γ_{0,x}^{ρ(τ)t}\left(\frac1{1 - τ} s\right) & \text{if $s ≤ 1 - τ$ and $τ {\lt} 1$}\\ γ_{1,x}^{ρ(1-τ)t}\left(\frac1τ \big (s - (1- τ)\big )\right) & \text{if $s ≥ 1 - τ$ and $τ {\gt} 0$}\\ \end{cases} \]

It is clear that if $s = 1 - τ$ then both branches agree and are equal to $β(x)$. Therefore it is easy to see that $δ$ is continuous at $(τ, x, t, s)$ except when $(τ,s)=(1,0)$ or $(τ,s)=(0,1)$.

To show the continuity for $(τ,s)=(1,0)$, let $K$ be a compact neighborhood of $x$ in $E$. Then $γ_0$ is uniformly continuous on the compact set $K × [0, 1] × 𝕊^1$, which means that $γ_{0,x'}^t$ tends uniformly to the constant function $s ↦ β(x)$ as $(x', t)$ tends to $(x, 0)$. This means that $γ_{0,x'}^{ρ(τ)t'}$ tends uniformly to the constant function $s ↦ β(x)$ as $(τ, x', t')$ tends to $(1, x, t)$. This means that $δ$ is continuous at $(τ,s)=(1,0)$ (it is clear that the other branch also tends to $β(x)$). The continuity at $(τ,s)=(0,1)$ is entirely analogous.

The beautiful observation motivating the above formula is why each $δ_{τ, x}^1$ surrounds $g(x)$. The key is that the image of $δ_{τ, x}^1$ contains the image of $γ_{0,x}^1$ when $τ ≤ 1/2$, and contains the image of $γ_{1,x}^1$ when $τ ≥ 1/2$. Hence $δ_{τ, x}^1$ always surrounds $g(x)$.

Corollary 1.15 ✓

Let $U_0$ and $U_1$ be open sets in $E$. Let $K_0 ⊆ U_0$ and $K_1 ⊆ U_1$ be compact subsets. For any $γ_0 ∈ \operatorname{\mathcal{L}}(U_0, g, β, Ω)$ and $γ_1 ∈ \operatorname{\mathcal{L}}(U_1, g, β, Ω)$, there exists $U ∈ 𝓝(K_0 ∪ K_1)$ and there exists $γ ∈ \operatorname{\mathcal{L}}(U, g, β, Ω)$ which coincides with $γ_0$ near $K_0\cup U_1^c$.

Proof ▼

Let $U_0'$ be an open neighborhood of $K_0$ whose closure $\overline U_0'$ is compact in $U_0$. Since $\overline U_0'$ and $K_1' := K_1 ∖ (K_1 ∩ U_0)$ are disjoint compact subsets of $E$, there is some continuous cut-off $ρ \! :E → [0, 1]$ which vanishes on $U_0'$ and equals one on some neighborhood $U_1'$ of $K_1'$.

Lemma 1.14 gives a homotopy of loops $γ_τ$ from $γ_0$ to $γ_1$ on $U_0 ∩ U_1$. On $U_0' ∪ (U_0 ∩ U_1) ∪ U_1'$, which is a neighborhood of $K_0 ∪ K_1$, we set

\[ γ_x = γ_{ρ(x), x} \]

which has the required properties.

Lemma 1.16

In the setup of Proposition 1.3, assume we have a continuous family $γ$ of loops defined near $K$ which is based at $β$, surrounds $g$ and such that each $γ_x^t$ takes values in $Ω_x$. Then there such a family which is defined on all of $U$ and agrees with $γ$ near $K$.

Proof ▼

Let $U_0$ be an open set containing $K$ such that $γ$ forms a surrounding family of loops on $U_0$. Let $U_i$, $i ≥ 1$ be a locally finite family of open sets with local surrounding families of loops $γ^i$ and compact subsets $K_i ⊂ U_i$ covering $\overline U$.

This is possible, since by Lemma 1.13 there is a local surrounding family of loops around each point in $\overline U$. Since $E$ is locally compact we may pick a compact neighborhood around each point in $\overline U$ with such a local family of loops. Since $\overline U$ is paracompact second countable, we can pick a countable refinement $U_i$ which is locally finite and still covers $\overline U$. By the shrinking lemma we can pick closed (and hence compact) subsets $K_i ⊂ U_i$ that also cover $\overline U$.

Now we define a family $(δ^i)_{i \in NN}$ by $δ^0 = γ|_{U_0}$ and $δ^{i+1}$ is obtained by extending $δ^i$ using $γ^i$ via Corollary 1.15. Since $δ^{i+1}$ equals $δ^i$ on $U_i^c$, this sequence is locally eventually constant. Therefore it has a well-defined limit $δ$ which is defined on continuous on all $U$. Since $δ^{i+1}$ equals $δ^i$ on a neighborhood of $K\cup \bigcup _{j{\lt}i} K_i,$ we know in particular that $δ^i$ equals $γ$ on a neighborhood of $K$. Since $K$ is compact, it has some neighborhood $O$ such that $δ^i|_{O}$ is eventually constant with eventual value $δ|_{O}$. Hence $δ=γ$ on a neighborhood of $K$.

1.3.2 The reparametrization lemma

The second ingredient needed to prove Proposition 1.3 is a parametric reparametrization lemma.

Lemma 1.17

Let $γ \! :E × 𝕊^1 → F$ be a smooth family of loops surrounding a map $g$ with base $β$ over some $U ⊆ E$. There is a family of circle diffeomorphisms $φ \! :U × 𝕊^1 → 𝕊^1$ such that each $γ_x ∘ φ_x$ has average $g(x)$ and $φ_x(0) = 0$.

Proof ▼

For any fixed $x$, since $γ_x$ strictly surrounds $g(x)$, there are points $s_1, …, s_{n+1}$ in $𝕊^1$ such that $g(x)$ is surrounded by the corresponding points $γ_x(s_j)$.

Let $μ_1, …, μ_{n+1}$ be smooth positive probability measures very close to the Dirac measures on $s_j$ (ie. $μ_j = f_j\, ds$ for some smooth positive function $f_j$ and, for any function $h$, $\int h\, dμ_j$ is almost $h(s_j)$). We set $p_j = \int γ_x\, d\mu _j$, which is almost $γ_x(s_j)$ so that $g(x) = \sum w_j p_j$ for some weights $w_j$ in the open interval $(0, 1)$ according to Lemma 1.9.

If $x'$ is in a sufficiently small neighborhood of $x$, Lemma 1.9 gives smooth weight functions $w_j$ such that $g(x') = \sum w_j(x')p_j(x')$. Let $U^i$, $i ≥ 1$ be a locally finite cover of $U$ by such neighborhoods, with corresponding measures $μ_j^i$, moving points $p_j^i$ and weight functions $w_j^i$. Let $(ρ_i)$ be a partition of unity associated to this covering. For every $x$, we set

\[ μ_x = \sum _{i=1}^∞ \sum _{j=1}^{n+1} ρ_i(x)w_j^i(x)\, μ_j^i \]

so that:

\begin{align*} \int γ_x\, dμ_x & = \sum _i ρ_i(x)\sum _{j=1}^{n+1} w_j^i(x) \int γ_x\, dμ_j^i\\ & = \sum _i ρ_i(x)\sum _{j=1}^{n+1} w_j^i(x) p_j^i(x)\\ & = \sum _i ρ_i(x) g(x) = g(x). \end{align*}

We now set $φ_x^{-1}(t) = \int _0^tdμ_x$ so that $g(x) = \overline{γ_x ∘ φ_x}$ for all $x$.

1.3.3 Proof of the loop construction proposition

We finally assemble the ingredients from the previous two sections.

Proof of Proposition 1.3 ▼

Let $γ^*$ be a family of loops surrounding the origin in $F$, constructed using Lemma 1.13. For $x$ in some neighborhood $U^*$ of $K$ where $g = β$, we set $γ_x = g(x) + εγ^*$ where $ε {\gt} 0$ is sufficiently small to ensure the image of $γ_x$ and its convex hull are contained in $Ω_x$ (recall $Ω$ is open and $K$ is compact). Lemma 1.16 extends this family to a continuous family of surrounding loops $γ_x$ for all $x$ (this is not yet our final $γ$).

We then need to approximate this continuous family by a smooth one. Some care is needed to ensure that it stays based at $β$. For instance, we can first compose each loop by some fixed surjective continuous map from $𝕊^1$ to itself that sends a neighborhood of $0$ to $0$. This way each loop becomes constant near $0$, and a convolution smoothing will then keep the value at $0$. If the smoothing is sufficiently $C^0$ small then the new $γ$ is still surrounding and takes values in $Ω$.

Then Lemma 1.17 gives a family of circle diffeomorphisms $h_x$ such that $γ^1_x ∘ h_x$ has average $g(x)$.

Finally we choose a cut-off function function $χ$ which vanishes on $\operatorname{Op}{K}$ and equals one on $\operatorname{Op}{U ∖ U^*}$. In $U^*$, we replace $γ_x ∘ h_x = g(x) + γ^* ∘ h_x$ by $g(x) + χ(x)γ^* ∘ h_x$. This operation does not change the average values of these loops, because it rescales them around their average value, but makes them constant on $\operatorname{Op}{K}$. Also, those loops stay in $Ω$, thanks to our choice of $ε$.