Errata for Axioms and Hulls (last updated 01 March 2025)

Page 4, line −4 ¬tpr … is a convex |tpr| ≤ 0 implies |pqr| > 0 by (1.2), so (1.3) expresses t as a convex Page 6, line 13 (1.4)and (1.4) and Page 6, line 19 pux, pvx, pwy, qvy, qwy, quz, rwz, ruz, rvx pux, pvz, pwy, qvy, qwx, quz, rwz, ruy, rvx Page 8, equation (2.6a) abd, dbc, acd, abd abd, dbc, acd, abc Page 12, line 3 str ∨ tpq str ∧ tpq Page 12, line −2 tems terms Page 14, line 18 Macmahon MacMahon Page 16, line 28 labelled labeled Page 20, line 10 (σ_{1} ∨ ⋯ ∨ σ_{1}) (σ_{1} ∨ ⋯ ∨ σ_{k}) Page 20, in (6.1) σ̄_{2} σ̄_{2} Page 21, line 1 cf_{j}, ′_{j}, c′′_{j}, c_{j}, c′_{j}, c′′_{j}, Page 22, line 10 q̄rs q̄rs̄ Page 23, lines 1 and 2 that when … we that t ∈ Δpqr implies t → p ∨ q → t, t → q ∨ r → t, and t → r ∨ p → t. Hence we Page 23, line 12 e^{iθj} e^{iθ_{j}} (twice) Page 23, line 15

[61] [68] Page 24, line 6

\begin{aligned}0 & → & 1 \\ ↑ & ↙ ↖ & ↑ \\ 4 & → & 6 \end{aligned}  \begin{aligned}0 & → & 1 \\ ↑ & ↙ ↖ & ↑ \\ 4 & ← & 6 \end{aligned}

Page 25, line 3 contains p → q contains r → p Page 29, line 1 adjaciences adjacencies Page 29, replacement for (8.1)

[figure]
bubble  cocktail-shaker  odd-even

(8.1)

Page 30, lines −5 and −2 (5.1) (5.2) Page 37, line 10 B^{n} B_{n} Page 37, line 16 can also be written leads to others such as Page 37, in (9.7) (α′′_{m} + 1) α′′_{m} Page 37, line −10 Equations Inequalities Page 39, line −15 The quantity … at least, this is Can the quantity 3^{n} in (9.8) be reduced to n2^{n−2}? This, at least, is Page 39, line −1 large. large. (And alas, the conjecture is false; see section 21.) Page 40, line −7 one circuit C′ on circuit C′′ Page 42, line −16 enough imply enough Page 42, line −10 960 1920 Page 46, line 10 paths points Page 46, line −5 (11.3) (11.4) Page 49, line −13 afer after Page 50, line 3 pp′t pp′t_{1} Page 50, line 6 t_{r}p′t_{r−1} t_{r−1}p′t_{r} Page 50, line 10 t_{3}t_{2}p′ t_{3}p′t_{2} Page 50, in (12.5) e_{<k} e_{<j} Page 51, line 3 x_{10} c_{10} Page 53, line 1 changed charged Page 57, in (14.4) (14.4) (14.4) Page 61, Step 3 x_{r} − x_{p} x_{r} = x_{p} [However, Lyle Ramshaw told me on 19 Nov 1993 that we can omit the clause ‘or x_{p} = x_{q} and y_{p} = y_{q} and x_{r} = x_{p} and y_{r} ≥ y_{p},’ since a valid CC system is obtained without this complication.] Page 64, replacement for the bottom 12 lines

if (x < y)

if (y < z) output (xyz)  x < y < z
else if (z < y)
if (x < z) output (xzy)  x < z < y
else output (zxy)  z ≤ x < y
else output (xzy)  x < y ≤ z ≤ y
else if (y < x)
if (x < z) output (yxz)  y < x < z
else if (y < z) output (yzx)  y < z ≤ x
else output (zyx)  z ≤ y < x
else if (x < z) output (xyz)  x ≤ y ≤ x < z
else output (zyx).  z ≤ x ≤ y ≤ x .

Page 72, line −11 same sign same sign as Page 73, line 9 ā, q, r, s} {ā, q, r, s} Page 73, line 24 allpoints all points Page 78, line −10 pt_{j}t_{j−1}t_{j−2} pt_{j−1}t_{j−2} Page 81, line −14 1000 n-gon 10000 n-gon Page 86, line −6 labelled labeled Page 87, line 8 p_{1}, …, p_{k} p_{1} … p_{k} Page 87, line 10 labelled labeled Page 88, line −13 p′_{1}, … p′_{r−2} p′_{1} … p′_{r−2} Page 91, line 8 fours form Page 91, line 23 t_{n} t_{m} Page 91, line 27 t_{m}t_{m−1}, …, t_{1} (t_{m}, t_{m−1}, …, t_{1}) Page 95, line 11 48, 50 48, 51 Page 97, lines 15 and 16 Laffaile Laffaille (twice) Page 97, new material just before section 22 Addendum, added March 2025 : A paper by Stephan Felsner and Pavel Valtr, “Coding and counting arrangements of pseudolines” [Discrete & Computational Geometry 46 (2011), 405–416] proves that the constant 3 in the theorem of section 9 can be improved to 2.487. It also mentions that Ondřej Bílka has disproved the conjecture following that theorem, by constructing reflection networks with Ω(2.076^{n}) cutpaths.

A paper by Xavier Goaoc and Emo Welzl, “Convex hulls of random order types” [Journal of the ACM 70 (2023), 8:1–8:47] shows among other things that the average number of points on the convex hull of a random n-point CC system is 4 − 8/(n^{2} − n + 2). Page 100, near the top I meant to cite also the paper by Franz Aurenhammer, “Voronoi diagrams — a survey of a fundamental geometric data structure,” ACM Computing Surveys 23 (1991), 345–405. Page 100, reference [5] (1992) (1993) Page 100, reference [7] in preparation. unpublished preprint. Page 100, reference [13] Krystallographie Kristallographie Page 101, reference [19] SIAM … 108–123. SIAM Journal on Computing 22 (1993), 418–429. Page 102, reference [32] 1992), to appear. 1993), 103–134. Page 102, reference [34] London Interscience London: Interscience Page 102, reference [37] Algorithmica, to appear. Algorithmica 9 (1993), 534–560. Page 103, reference [45] Addison-Wesley Addison–Wesley Page 103, reference [47] , book in preparation. (New York: ACM Press, 1994), viii + 576 pp. Page 103, reference [54] Discrete … 1990. Syomposium on Computational Geometry 6 (1990), 16–22. Page 104, reference [59] 331-332 331–332 Page 104, reference [60] probléme problème Page 104, reference [61] realisierbarkeitskriterien Realisierbarkeitskriterien Giessen Gießen Page 104, reference [69] 133 (1907) 133 (1908) Page 104, reference [71] Eidgenossische Eidgenössische Page 105, new entry Aurenhammer, Franz, 100. Page 105, new entry Bílka, Ondřej, 97. Page 105, entry for Bokowski 95, 96 95–97 Page 106, left column, line 1 vii vii. Page 106, left column Dress, Andreas Dress, Andreas Walter Martin Page 106, new entry Felsner, Stephan, 97. Page 106, new entry Goaoc, Xavier, 97. Page 107, entry for Lawrence 95 95–96 Page 107, left column Macmahon MacMahon Page 108, entry for perturbations 59-60 59–60 Page 109, top line of left column instructions! 74 instructions, 74 Page 109, new entry Valtr, Pavel, 97. Page 109, entry for Welzl 103 97, 103