Geometric Proofs of Some Recent Results of Yang Lu

The average‐case area of Heilbronn‐type triangles*

“…Heilbronn ( -1975 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].…”

“…"2 log i" and not "log i" since we need to encode i in such a way that U can determine the end of the encoding. 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.…”

Constructing a Tetrahedron with Prescribed Heights and Widths

“…Now, let us discuss an nontypical geometric constraint problem. Given are a set of constraints on four points P 1 , P 2 , P 3 , P 4 in E 3 : the distances from each point to the plane determined by the other three points, namely, d(P 1 , P 2 P 3 P 4 ), d(P 2 , P 3 P 4 P 1 ), d(P 3 , P 4 P 1 P 2 ), d(P 4 , P 1 P 2 P 3 ), m m m m m m Since any tetrahedron has 6 freedoms up to isometries, it is possible to establish a formula connecting the seven quantities h 1 , h 2 , h 3 , h 4 , τ 12 , τ 13 , τ 14 The following, which was proved repeatedly in literatures (see [2] and [10] for the history and proofs), gives a very simple equality.…”