Read-back: Hirsch.santos_counterexample

The theorem asserts the existence of natural numbers $d$ and $n$ (both existentially quantified, with no lower bounds — $d = 0$ or $n = 0$ are not excluded by the statement itself), together with a family of vectors $a_1, \dots, a_n \in \mathbb{R}^d$ (Euclidean space with the standard inner product) and real numbers $b_1, \dots, b_n \in \mathbb{R}$, such that the set

(the solution set of the $n$ non-strict linear inequalities; when $n = 0$ this would be all of $\mathbb{R}^d$) satisfies all three of the following:

1. $P$ is nonempty.

2. $P$ is bounded (in the sense of the metric/bornology of $\mathbb{R}^d$).

3. The "diameter at most $n - d$" predicate fails for $P$, where $n - d$ is truncated natural-number subtraction (equal to $0$ whenever $d \ge n$). Spelled out, the predicate that is being negated says: for every pair of extreme points $u, v$ of $P$ (extreme points of $P$ as a subset of $\mathbb{R}^d$, i.e. points of $P$ that do not lie strictly inside any segment whose endpoints are in $P$), there exists a sequence of points $w_0, w_1, w_2, \dots \in \mathbb{R}^d$ (indexed by all of $\mathbb{N}$, with only the first $n - d + 1$ terms constrained) such that

   • $w_0 = u$ and $w_{\,n-d} = v$, and
   • for each index $i < n - d$, either $w_i = w_{i+1}$ (a stationary step), or $w_i$ and $w_{i+1}$ are "adjacent on $P$", meaning $w_i \ne w_{i+1}$ and the closed segment $[w_i, w_{i+1}]$ is an extreme subset of $P$ (i.e. $[w_i, w_{i+1}] \subseteq P$, and whenever a point of $[w_i, w_{i+1}]$ lies in the open segment between two points of $P$, both of those points already belong to $[w_i, w_{i+1}]$).

   The negation (item 3) therefore asserts: there exist extreme points $u, v$ of $P$ for which no such sequence of $n - d$ steps (each step stationary or crossing such an "adjacent" pair) leads from $u$ to $v$.

Note that the intermediate points $w_i$ of a walk are quantified over all of $\mathbb{R}^d$; they are only forced to relate to $P$ through the adjacency condition on non-stationary steps (a stationary step imposes no membership in $P$). Also note that if $P$ had at most one extreme point, or none, the diameter predicate would hold vacuously, so item 3 implicitly forces $P$ to have at least two distinct extreme points. The statement does not require the walk length bound $n - d$ to be positive, does not require $n > d$, does not require the polytope to be full-dimensional or simple, and does not assert anything about which specific $d$ and $n$ witness the claim.