Tightness conjecture for the fat-shattering dimension of bounded affine maxima

Let $B ewcommand{\mathbb{R}}{\mathbb{R}}\subset\mathbb{R}^d$ be the Euclidean unit ball. For $i\in[k]$, let

$$F_i=\left\{x\mapsto w\cdot x+b:\ \left\Vert w\right\Vert\vee|b|\le R_i\right\}$$

be collections of bounded affine functions on $B$, and let $F_{\max}$ denote their $k$-fold maximum. Theorem~ states that

$$\operatorname{fat}_{\gamma}(F_{\max})\le \frac{c\operatorname{Log}(k)}{\gamma^2}\sum_{i=1}^kR_i^2, \qquad 0<\gamma<\min_{i\in[k]}R_i,$$

for a universal constant $c>0$. Tightness conjecture. The bound in Theorem~ is tight, in the sense that it has a matching lower bound. This would establish optimality of the logarithmic dependence on $k$ for bounded affine function classes; the cited upper bound is presented as sharper than earlier work, while the matching lower bound remains unstated and open here.