Introduction

A norm may force the average distance to two opposite points to grow in every sufficiently large direction, yet have no equivalent norm with the corresponding one-sided growth. We construct two explicit real \(L^1\) spaces that exhibit this distinction. In both spaces, multiplying along a tree has two effects: fresh variables force midpoint growth, while path products of \(L^1\) norm one prevent asymptotically uniformly convex renormings.

For a norm \(N\) on an infinite-dimensional real Banach space \(X\), let \(\operatorname{cof}(X)\) denote the closed linear subspaces of finite codimension. For \(t>0\), define the averaged asymptotic midpoint modulus and the usual one-sided asymptotic modulus (Johnson et al. 2002, sec. 2.1) by \[\begin{align*} \widehat\delta_N(t) &=\inf_{N(x)=1}\sup_{F\in\operatorname{cof}(X)} \inf_{\substack{y\in F\\N(y)\ge1}} \left(\frac{N(x+ty)+N(x-ty)}2-1\right),\\ \overline\delta_N(t) &=\inf_{N(x)=1}\sup_{F\in\operatorname{cof}(X)} \inf_{\substack{y\in F\\N(y)=1}}\bigl(N(x+ty)-1\bigr). \end{align*}\] The norm is asymptotically uniformly convex (AUC) if \(\overline\delta_N(t)>0\) for every \(t>0\). The subspace \(F\) may depend on the center \(x\) and on \(t\); it must work for all the indicated directions. The standard asymptotic midpoint modulus uses unit directions \(N(y)=1\) and replaces the displayed average by the maximum of the two norms. A norm is asymptotically midpoint uniformly convex (AMUC) when this maximum modulus is positive for every \(t>0\) (Dilworth et al. 2016, Definition 2.2). All numerical midpoint estimates below concern the displayed average. They imply the same lower bounds for the maximum, and hence AMUC, because the maximum dominates the average.

Let \(\mathcal T=\mathbb N^{<\omega}\) be the tree of finite sequences of positive integers, with root \(\varnothing\). For a nonroot vertex \(v\), write \(v^-\) for its parent. Fix a positive nonconstant real random variable \(W\) such that \[\mathbb EW=1,\qquad m_2:=\mathbb EW^2<\infty.\] On a countable product probability space take independent copies \(W_v\), indexed by the nonroot vertices, and define \[P_{\varnothing}=1,\qquad P_v=P_{v^-}W_v, \qquad X_W= \overline{\operatorname{span}_{\mathbb R}\{P_v:v\in\mathcal T\}}^{\,L^1}.\] We use the inherited real \(L^1\) norm, denoted by \(\lVert \cdot\rVert_1\). Write \(X_{\mathrm{exp}}\) for the law with density \(e^{-w}\) on \((0,\infty)\), and \(X_{\mathrm{sq}}\) for the law of \(G^2\), where \(G\) is a standard real Gaussian. For \(K>0\) set \[c(K)=\frac1{16}\mathbb E(|G|-2K)_+,\qquad r_+=\max\{r,0\}.\]

The three estimates come from different scalar arguments, whose parameter-dependent forms are proved below. None of the displayed constants is asserted to be optimal. The general multiplier hypothesis is used for the renorming obstruction; the positive midpoint estimates are proved for the two specified laws.

Finite conditional kernels and fresh multipliers

We first construct the subspaces of finite codimension needed for the midpoint estimates. The point is that conditioning on finitely many coordinates has finite rank on the actual closed span \(X_W\).

Independence and positivity give \[ \lVert P_v\rVert_1=1,\qquad \mathbb EP_v^2=m_2^{|v|}. \tag{4}\] As a closed subspace of \(L^1\), \(X_W\) is a Banach space. The first-generation differences \(P_{(n)}-1=W_{(n)}-1\) are nonzero and pairwise orthogonal in \(L^2\), so every finite subset is linearly independent, also as elements of \(L^1\). Thus \(X_W\) is infinite dimensional. The first-generation products themselves are also linearly independent: taking expectations in a relation \(\sum_n a_nW_{(n)}=0\) gives \(\sum_n a_n=0\), reducing it to the corresponding relation among the orthogonal differences. Rational finite combinations of the countable family of products are dense, so \(X_W\) is separable.

Let \(T\subset\mathcal T\) be finite, contain the root, and contain every predecessor of each of its vertices. Write \(\mathcal G_T\) for the \(\sigma\)-field generated by the coordinates \(W_v\) with \(v\in T\setminus\{\varnothing\}\). For \(v\in\mathcal T\), let \(v_T\) be its last ancestor in \(T\). Conditional expectation satisfies \[ Q_TP_v:=\mathbb E(P_v\mid\mathcal G_T)=P_{v_T}. \tag{5}\] Consequently \(Q_T\) restricts to a contractive projection of \(X_W\) onto the finite-dimensional space \[E_T=\operatorname{span}\{P_v:v\in T\}, \qquad F_T=\ker(Q_T|_{X_W})\in\operatorname{cof}(X_W).\] This assertion on the completion follows from the continuity of conditional expectation and the closedness of \(E_T\). Applying \(I-Q_T\) to finite-product approximations shows that finite-product combinations in \(F_T\) are dense in \(F_T\).

The same space and tails can also be described using increments. Put \(u_v=P_v-P_{v^-}\) for \(v\ne\varnothing\). Telescoping along paths shows that \(\{1,u_v:v\ne\varnothing\}\) and \(\{P_v:v\in\mathcal T\}\) have the same algebraic span. Formula (5) gives \(Q_Tu_v=u_v\) for \(v\in T\) and \(Q_Tu_v=0\) for \(v\notin T\). Applying \(I-Q_T\) to the algebraic span and then passing to its closure proves \[ F_T=\overline{\operatorname{span}\{u_v:v\notin T\}}^{\,L^1}. \tag{6}\]

For \(t>0\) and \(x,y\in X_W\), define the midpoint increment \[M_t(x,y)=\frac{\lVert x+ty\rVert_1+\lVert x-ty\rVert_1}2-\lVert x\rVert_1.\] The real scalar identity \((|a+b|+|a-b|)/2=\max\{|a|,|b|\}\) gives \[ M_t(x,y)=\mathbb E(t|y|-|x|)_+. \tag{7}\] For \(u\in E_T\) and \(y\in F_T\), we seek a lower bound on this excess in terms of \(\lVert y\rVert_1\) and \(\lVert u\rVert_1\), uniform over \(T\). Since \(u\) is \(\mathcal G_T\)-measurable, we first separate the fresh multipliers in \(y\) from the coordinates determining \(u\).

Now write a finite-product combination in the kernel as \(y=\sum_{v\in S}\alpha_vP_v\in F_T\), where \(S\subset\mathcal T\) is finite. Write \(w\preceq v\) when \(w\) is an ancestor of \(v\), including \(v\) itself, and use \(w\prec v\) for strict ancestry. Let \(I\) be the finite set of first vertices outside \(T\) on paths to vertices of \(S\setminus T\). Factoring out the multiplier at each such exit gives \[d=\sum_{v\in S\cap T}\alpha_vP_v,\qquad B_i=P_{i^-}\sum_{\substack{v\in S\\i\preceq v}} \alpha_v\prod_{i\prec w\preceq v}W_w\quad(i\in I),\] where an empty product is \(1\). Thus \(d\) is \(\mathcal G_T\)-measurable, and \(B_i\) depends only on old coordinates and on strict descendants of \(i\), not on \(W_i\). Put \(m_i=\mathbb E(B_i\mid\mathcal G_T)\). The factorization and the condition \(Q_Ty=0\) yield \[ y=d+\sum_{i\in I}W_iB_i,\qquad d=-\sum_{i\in I}m_i, \qquad \sigma=\Big(\sum_{i\in I}B_i^2\Big)^{1/2}. \tag{8}\] Distinct first-exit vertices have disjoint descendant subtrees. Hence, conditionally on \(\mathcal G_T\), the \(B_i\) are independent, and their entire vector is independent of the first-exit variables \(W_i\). In particular, conditioning further on all \(B_i\) leaves the \(W_i\) independent with their original law. Write \[\mathcal H_T=\mathcal G_T\vee\sigma(B_i:i).\] We will condition on \(\mathcal G_T\) when copying entire coefficients, and on \(\mathcal H_T\) when resampling only first-exit multipliers. Figure 1 illustrates this distinction. All quantities used here are integrable; in fact the finite products belong to \(L^2\).

The next two estimates measure the size of a tail vector by the expected Euclidean size of its coefficient list. Their upper bounds on \(\lVert y\rVert_1\) give lower bounds on \(\mathbb E\sigma\), which the scalar estimates in Sections 3 and 4 will convert into midpoint gains. They use independent-copy symmetrization followed by averaging over independent signs; this is the standard symmetrization method, here applied conditionally on the old coordinates (Bartlett and Mendelson 2002, proof of Theorem 8, p. 468). Both constants will be useful: they come from different ways of centering the sum.

The preceding construction applies to finite-product tails. To pass midpoint estimates to their closure, the triangle inequality gives \[ |M_t(x,y)-M_t(x',y')| \le2\lVert x-x'\rVert_1+t\lVert y-y'\rVert_1. \tag{11}\] Thus all tail estimates below pass to the completed tail subspace before we impose the condition \(\lVert y\rVert_1\ge1\).

The lemma is applied to norm inequalities, not to an infinite first-exit expansion. Thus no representation or extra integrability of a general completed tail is needed. It also fixes the order of choices: \(t\) and \(x\) determine \(T\) before the direction \(y\) is allowed to vary.

Exponential products: a uniform excess estimate

The exponential law supplies positive expected excess beyond every multiple of the coefficient size. A Gaussian scale mixture makes that excess uniform over all real coefficient lists and all shifts. Take \(W=Z\) to have the exponential distribution with density \(e^{-z}\) on \((0,\infty)\), and write \(X_{\mathrm{exp}}=X_Z\). Here \(m_2=2\), so Lemma 2 gives \(\lVert y\rVert_1\le3\mathbb E\sigma\). Let \(G\) denote a standard real Gaussian and put, for \(K>0\), \[ c(K)=\frac1{16}\mathbb E(|G|-2K)_+>0. \tag{12}\] The Laplace distribution as a Gaussian scale mixture belongs to the theory of normal scale mixtures (Andrews and Mallows 1974); the characteristic-function calculation below proves the exact identity needed here. We prove the uniform conditional bound \[ \mathbb E\bigl((|y|-K\sigma)_+\mid\mathcal G_T,(B_i)_i\bigr) \ge c(K)\sigma. \tag{13}\] After conditioning as indicated, all coefficients, \(d\), and \(\sigma\) are fixed. Resample only the exponential multipliers to obtain \(y'\) with the same conditional law. The triangle inequality and the positive-part inequality give \[2\mathbb E(|y|-K\sigma)_+ \ge\mathbb E(|y-y'|-2K\sigma)_+.\] The difference of two independent standard exponentials has characteristic function \((1+q^2)^{-1}\) at \(q\in\mathbb R\). This also equals \(\mathbb E\exp(-q^2U)\) for a standard exponential \(U\), so that difference has the law of \(\sqrt{2U}\,G\), with \(U\) and \(G\) independent. Using independent such representations for the different multipliers, if \(\sigma>0\) the law of \((y-y')/\sigma\) is a centered Gaussian mixture with variance \[V=\frac{2\sum_iB_i^2U_i}{\sigma^2},\qquad \mathbb EV=2,\qquad \mathbb EV^2=4\left(1+\frac{\sum_iB_i^4}{\sigma^4}\right)\le8.\] Since \(\mathbb E[V\mathbf1_{\{V\ge1\}}]\ge1\), Cauchy–Schwarz gives \(\mathbb P(V\ge1)\ge1/8\). On that event the Gaussian excess above \(2K\) is at least \(\mathbb E(|G|-2K)_+\). Hence \[\mathbb E(|y-y'|-2K\sigma)_+ \ge\frac{\sigma}{8}\mathbb E(|G|-2K)_+,\] which proves (13). For \(\sigma=0\) the asserted lower bound is zero, so the conclusion holds without normalization.

Gaussian squares: two scalar mechanisms

Now take \(W=G^2\), where \(G\) is standard Gaussian, and put \(X_{\mathrm{sq}}=X_{G^2}\). Here \(m_2=3\). In particular, the product and jump descriptions above give the same closed subspace with the same norm. We first establish the Gaussian difference calculation that both estimates will use, while keeping their subsequent probability and convexity steps separate.

For fixed real coefficients \(a_1,\ldots,a_k\) and a fixed \(b\in\mathbb R\), put \[H=b+\sum_i a_i(G_i^2-1),\qquad s=\Big(\sum_i a_i^2\Big)^{1/2},\] where the \(G_i\) are independent standard Gaussians. Let \(H'\) be an independent copy with the same coefficients and shift. The Gaussian polarization identity used in decoupling (Levina and Vershynin 2012, proof of Lemma 3.2) is especially simple here. The orthogonal change of variables \(A_i=(G_i+G_i')/\sqrt2\), \(C_i=(G_i-G_i')/\sqrt2\) shows that \[H-H'\ \text{has law}\ 2\sum_i a_iA_iC_i,\] where all \(A_i,C_i\) are independent standard Gaussians. Conditionally on the \(A_i\), this is Gaussian with variance \(4V\), where \[V=\sum_i a_i^2A_i^2,\qquad \mathbb EV=s^2,\qquad \mathbb EV^2=s^4+2\sum_i a_i^4\le3s^4.\] This is the elementary second-moment step usually called the Paley–Zygmund inequality (Paley and Zygmund 1932b, 1932a). We include the argument: for \(s>0\), the inequality \(\mathbb E[V\mathbf1_{\{V\ge s^2/2\}}]\ge s^2/2\) and Cauchy–Schwarz yield \(\mathbb P(V\ge s^2/2)\ge1/12\). On this event the conditional standard deviation of \(H-H'\) is at least \(\sqrt2s\). Consequently, for every \(K>0\), \[ \mathbb P(|H-H'|\ge2Ks) \ge q_K,\qquad q_K=\frac{\mathbb P(|G|\ge\sqrt2K)}{12}>0. \tag{15}\] The event in (15) has probability one when \(s=0\), so the same statement remains valid in that case.

The two proofs control different scalar quantities. Proposition 6 first estimates a shifted original variable and pays a factor of two in the union bound. Proposition 7 estimates a difference and uses even convexity directly in the midpoint increment, preserving the denominator \(12\). Their first-moment constants are respectively \(2+\sqrt2\) and \(2\sqrt3\). Neither proof requires the original shifted variable to be symmetric.

Bounded paths exclude AUC renormings

The midpoint arguments depend on the multiplier distribution. Here we use only positivity, a nonconstant mean-one law, and a finite second moment to rule out AUC renormings. At every vertex the child increments are weakly null and have the same positive norm. An AUC norm would therefore grow by a fixed factor at each step of a suitably chosen branch, whereas all path products have inherited norm one.

For each fixed vertex \(v\) and its children \(v^\frown n\), put \[D_{v,n}=P_{v^\frown n}-P_v=P_v(W_{v^\frown n}-1).\] Independence and (4) give \[ \lVert D_{v,n}\rVert_1=\rho:=\mathbb E|W-1|>0,\qquad \mathbb ED_{v,n}^2=m_2^{|v|}(m_2-1). \tag{22}\] For fixed \(v\) the sequence \((D_{v,n})_n\) is pairwise orthogonal in \(L^2\) and has constant finite \(L^2\) norm. Bessel’s inequality therefore makes it weakly null in \(L^2\). The inclusion \(L^2\longrightarrow L^1\) is continuous on this probability space. Moreover, each continuous linear functional on \(X_W\) extends to \(L^1\) by Hahn–Banach and hence restricts to a continuous functional on \(L^2\). Thus \((D_{v,n})_n\) is weakly null in \(X_W\) itself.

Suppose \(N\) is equivalent to the inherited norm, with \(a\lVert z\rVert_1\le N(z)\le b\lVert z\rVert_1\) for \(0<a\le b<\infty\). We will show the precise obstruction \[ \overline\delta_N\left(\frac{a\rho}{2b}\right)=0. \tag{23}\] The usual one-sided modulus is nonnegative: for each unit center \(x\), a norming functional gives a finite-codimensional kernel on which \(N(x+ty)\ge1\). It remains to rule out positivity at the displayed scale.

Assume, to the contrary, that \[t_0=\frac{a\rho}{2b},\qquad \delta=\overline\delta_N(t_0)>0,\qquad \gamma=\delta/2.\] At a vertex \(v\), we have \(a\le N(P_v)\le b\). The modulus at the unit center \(P_v/N(P_v)\) supplies a closed finite-codimensional subspace \(F\) such that \[N(P_v+z)\ge(1+\gamma)N(P_v) \quad(z\in F,\ N(z)\ge t_0N(P_v)).\] Indeed, its definition first gives the estimate at equality of the radius. For \(N(x)=N(w)=1\), with \(w\in F\) and \(r\ge t_0\), convexity yields \[N(x+rw)-1\ge\frac r{t_0}\bigl(N(x+t_0w)-1\bigr),\] so the estimate extends to all larger radii.

Equivalent norms induce the same weak topology. The quotient images of the weakly null sequence \(D_{v,n}\) in the finite-dimensional space \(X_W/F\) therefore tend to zero in quotient norm. We may choose \(z_n\in F\) with \(N(D_{v,n}-z_n)\to0\). Eventually \[N(z_n)\ge a\rho/2=t_0b\ge t_0N(P_v).\] Choose such an \(n\) with approximation error at most \(\gamma N(P_v)/2\). Then \[N(P_{v^\frown n}) \ge N(P_v+z_n)-N(D_{v,n}-z_n) \ge(1+\gamma/2)N(P_v).\] The same \(\gamma\) works at every vertex, although \(F\) and \(n\) may change. Repeatedly selecting a child gives a depth-\(k\) product with norm at least \(a(1+\delta/4)^k\), contrary to the uniform upper bound \(b\). This proves (23) and the renorming assertion of Theorem 1.

Finally \(0<\rho<2\): nonconstancy gives positivity, and \(\mathbb E|W-1|=2-2\mathbb E\min\{W,1\}<2\) because \(W>0\) almost surely. Thus \(0<t_0<1\), so positivity of the one-sided modulus on \((0,1)\) alone would already contradict the tree. For exponential multipliers \(\rho=2/e\); indeed \(\mathbb E|Z-1|=2\mathbb E(Z-1)_+=2\int_1^\infty e^{-s}\,ds\). The weak-null calculation requires only a finite \(L^2\) bound at each fixed parent, not a bound uniform over the depth.