Abstract: In this paper, we show
lower and upper bounds for a generalization of Heilbronn’s triangle problem to
d dimensions. Namely, we show that
there exists a set (resp., of n
points in the d-dimensional unit cube such that the minimum-area triangle
(embedded in d dimensions) defined by
some three points of (resp., has an area of (resp., We then generalize the applied
methods and show that there exists a set (resp., of n
points in the d-dimensional unit cube such that the minimum-volumek-dimensional simplex (embedded
in d dimensions, for defined by some points of (resp., has volume where is
independent of n (resp.,
Keywords and phrases: Heilbronn’s triangle problem, probabilistic method.
