Geometric Proofs of Some Recent Results of Yang Lu

The average‐case area of Heilbronn‐type triangles*

“…One way to do that is to use the code 1 l (l(i))0l(i)i which has length 2l(l(i)) + l(i) + 1 < 2 log i bits. 4 In what follows, "log" denotes the binary logarithm. "⌊r⌋" is the greatest integer q such that q ≤ r.…”

“…Heilbronn asked for the exact value or approximation of ∆ n . The list [1,2,3,4,7,8,13,14,17,18,19,20,21,22,23,24,25,26,27] is a selection of papers dealing with the problem. Obviously, the value of ∆ n will change only by a small constant factor for every unit area convex shape, and it has become customary to consider the unit square [22].…”

The expected size of Heilbronn's triangles

“…:= v 2 ; ,v 1 orthogonal to the vector v , u with the vector w , u : = w 1 ; w 2 equals the product of their lengths times the cosine of the enclosed angle. The real Euclidean length of the vector v 2 ; ,v 1 i s l while the real Euclidean length of vector w 1 ; w 2 times the cosine of the enclosed angle equals h. Therefore, the inner product is two times the area of the triangle u; v; w and we h a ve 2 K 2 = jv 2 w 1 , v 1 w 2 j, which is necessarily a non-zero multiple of g. 5 Given the long side u; v w e can determine the numberg of grid points incident o n u; v. Since 2K 2 i s a m ultiple of g, w e know that 2K 2 = f g where f is integer.…”

“…The concatenated descriptions of the ordered list of coordinates of x 1 ; : : : ; x n with those of w deleted in log , K 2 n,1 bits; 5 The quantity 2 K 2 equals the number of grid points contained in one of the two rectangles in Figure 1, possibly up to an additional term of order of circumference of the rectangle. Intuitively, w e can describe w by giving its index in an enumeration of the set of these grid points.…”