No abstract

We consider the problem of bounding the maximum possible number f k,d (n) of ksimplices that are spanned by a set of n points in R d and are similar to a given simplex. We first show that f 2,3 (n) = O(n 13/6 ), and then tackle the general case, and show

Abstract. We show that the maximum number of occurrences of a given angle in a set of n points in R 3 is O(n 7/3 ), and that a right angle can actually occur Ω(n 7/3 ) times. We then show that the maximum number of occurrences of any angle different from π/2 in a set of n points in R 4is O(n 5/2 β(n)), where β(n) = 2 O(α(n) 2 ) and α(n) is the inverse Ackermann function.

Let P be a set of n points in R 3 , and k ≤ n an integer. A sphere σ is k-rich with respect to P if |σ ∩ P | ≥ k, and is η-nondegenerate, for a fixed fraction 0 < η < 1, if no circle γ ⊂ σ contains more than η|σ ∩ P | points of P .We improve the previous bound given in [1] on the number of k-rich η-nondegenerate spheres in 3-space with respect to any set of n points in R 3

We show that the number of unit-area triangles determined by a set of n points in the plane is O(n 9/4+ε ), for any ε > 0, improving the recent bound O(n 44/19 ) of Dumitrescu et al.