Gruber P. Convex and Discrete Geometry

An Irregularity Criterion
13 The Space of Convex Bodies 233
If an approximation or iteration procedure is very fast or very slow for a dense set of
elements of a space, what can be said for typical elements? The following result of
Gruber [418] gives a Baire type answer.
Theorem 13.2. Let B be a Baire space. Then the following statements hold:
(i) Let α1,α2, ··· > 0 and let f1, f2, ··· : B →[0, +∞) be continuous functions
such that the set � x ∈ B : fn(x) = o(αn) as n →∞ � is dense in B. Then for
most x ∈ B the inequality fn(x) 0 and let g1, g2, ··· : B →[0, +∞) be continuous functions
such that the set � x ∈ B : βn = o � gn(x) � as n →∞ � is dense in B. Then for
most x ∈ B the inequality βn < gn(x) holds for infinitely many n.
Proof. (i) Since the functions fn are continuous, the sets � �
x ∈ B : fn(x) ≥ αn are
closed. Hence
Bn = � x ∈ B : fn(x) ≥ αn, fn+1(x) ≥ αn+1,... � is closed.
The assumption in (i) implies that int Bn = ∅. Thus, Bn is nowhere dense and
therefore,
∞�
Bn = � x ∈ B : fn(x) ≥ αn for all but finitely many n � is meagre.
n=1
This implies (i) on noting that
B\
∞�
Bn = � x ∈ B : fn(x)