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Abstract. In this paper we prove four new (infinite) lists of quadratic inequalities, and four cubic inequalities, for the flag f -vectors of 4-polytopes. These extend and supplement the only four currently known non-linear inequalities, which were proved by Bayer in 1987. The new lists of inequalities for flag f -vectors yield new lists of inequalities for f -vectors of 4-polytopes. Using the latter, we managed to improve an estimate discovered by Höppner and Ziegler concerning upper bounds of f 1 in terms of f 0 and f 3 .

Recently, Dekster introduced a new angle measure for Minkowski spaces according to which the total angular measure τ around a point in a two-dimensional Minkowski space need not be 2π. In this paper, we shall show that while τ need not be 2π, τ always lies between √ 2π and 8. (2000): Primary 51B20, 52A38. Mathematics Subject ClassificationLet B be a centrally symmetric compact convex subset of E 2 with nonempty interior, and let M denote the two-dimensional Minkowski space for which B is the closed unit disk.Dekster's angle measure in M, defined in [1], is given bywhere ϕ denotes the (oriented) Euclidean angles between the positive x−axis and the ray R ϕ with initial point O, r (ϕ) is the Euclidean distance between O and the point P ϕ on the boundary of B lying along the ray R ϕ , and b (ϕ) equals the Euclidean distance between the two supporting lines parallel to the ray R ϕ . It is immediate that for every ϕ, 1. P ϕ is the point (r(ϕ) cos ϕ, r(ϕ) sin ϕ). We have4. If B is the Euclidean unit disk, then r (ϕ) = 1 and b (ϕ) = 2 for all ϕ, and so M (α, β) = β − α. 72

Purpose To compare postoperative anterior chamber inflammation, pain, and patient preference following small incision lenticule extraction (SMILE) in eyes treated with a dexamethasone 0.4 mg intracanalicular insert (DEX) or topical prednisolone acetate (PRED). Patients and Methods In this prospective, randomized, fellow eye-controlled trial, 20 patients underwent same-day, bilateral SMILE. One randomly-selected eye of each patient received DEX placed immediately postoperatively, and the fellow eye received topical PRED tapered over 2 weeks. Postoperative evaluations were performed on day 1, week 1, month 1, and month 3. Primary outcomes included postoperative pain, incidence of anterior chamber cell and flare, and patient preference of steroid therapy. Results No eyes in either group had any clinically evident cell or flare at any postoperative time point. Mean pain scores (0–10 by subjective report) and incidence of any pain were statistically similar at all postoperative visits. Uncorrected distance visual acuity improved in all eyes, 91% of which achieved 20/25 or better. No eyes lost any lines of corrected distance visual acuity. Three eyes developed a steroid-related rise in intraocular pressure, all of which resolved with 2 of the 3 eyes requiring topical therapy. At 1 week, 1 month, and 3 months, 70%, 65%, and 53% of patients preferred DEX over PRED therapy, respectively. Conclusion The DEX insert was preferred by more patients and controlled postoperative inflammation and pain comparably to topical PRED in eyes undergoing SMILE. There were no statistically significant differences in visual outcomes between the two groups.

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