Extreme points of the distance function on convex surfaces

Abstract

Abstract. We first see that, in the sense of Baire categories, many convex surfaces have quite large cut loci and infinitely many relative maxima of the distance function from a point. Then we find that, on any convex surface, all these extreme points lie on a single subtree of the cut locus, with at most three endpoints. Finally, we confirm (both in the sense of measure and in the sense of Baire categories) Steinhaus' conjecture that "almost all" points admit a single farthest point on the surface.

Farthest points on most Alexandrov surfaces

“…T. Zamfirescu strengthened Theorem 2 in [23] by proving that on any convex surface S, the set of those x ∈ S for which Fx contains more than one point is σ-porous [25]. This statement is not true for all Alexandrov surfaces, as one can easily see for the standard projective plane, but it would be interesting to prove it for most surfaces A ∈ B (κ).…”

“…We generalize results about convex surfaces from [23] by showing that, on most surfaces A ∈ B (κ), most points x ∈ A have a unique farthest point (Theorem 1) which is joined to x by precisely 3 segments (Theorem 2). In particular, Theorem 1 gives an answer to the last open problem in [25].…”

“…The study of farthest points on convex surfaces originated from some questions of H. Steinhaus, presented by H. T. Croft, K. J. Falconer and R. K. Guy in the chapter A35 of their book [4]. The questions have been answered since then, mainly by T. Zamfirescu in a series of papers [22], [23], [24], [25], and yielded several results about farthest points on most convex surfaces; see the survey [18]. A few results have also been obtained for general Alexandrov surfaces [19], [21].…”

Sets of tetrahedra, defined by maxima of distance functions

“…The next lemma follows from Lemma 3.4 and a more general result in [21]. Lemma 3.8 Let S be a convex surface and x a point in S. Then M x is contained in a minimal (by inclusion) Y -subtree of C(x), possibly degenerated to an arc or a point.…”

With respect to whom are you critical?

“…Or, at least, is it true for densely many surfaces homeomorphic to the sphere? For a similar -still open -problem concerning convex surfaces, see [23]. tudor.zamfirescu@mathematik.tu-dortmund.de…”