7 research outputs found
Dowker-type theorems for hyperconvex discs
A hyperconvex disc of radius r is a planar set with nonempty interior that is the intersection of closed circular discs of radius r . A convex disc-polygon of radius r is a set with nonempty interior that is the intersection of a finite number of closed circular discs of radius r . We prove that the maximum area and perimeter of convex disc- n -gons of radius r contained in a hyperconvex disc of radius r are concave functions of n , and the minimum area and perimeter of disc- n -gons of radius r containing a hyperconvex disc of radius r are convex functions of n . We also consider hyperbolic and spherical versions of these statements
On a Dowker-type problem for convex disks with almost constant curvature
A classical result of Dowker (Bull. Amer. Math. Soc. 50: 120-122, 1944)
states that for any plane convex body , the areas of the maximum (resp.
minimum) area convex -gons inscribed (resp. circumscribed) in is a
concave (resp. convex) sequence. It is known that this theorem remains true if
we replace area by perimeter, or convex -gons by disk--gons, obtained as
the intersection of closed Euclidean unit disks. It has been proved
recently that if is the unit disk of a normed plane, then the same
properties hold for the area of --gons circumscribed about a -convex
disk and for the perimeters of --gons inscribed or circumscribed
about a -convex disk , but for a typical origin-symmetric convex disk
with respect to Hausdorff distance, there is a -convex disk such that
the sequence of the areas of the maximum area --gons inscribed in is
not concave. The aim of this paper is to investigate this question if we
replace the topology induced by Hausdorff distance with a topology induced by
the surface area measure of the boundary of .Comment: 12 pages, 3 figure
Dowker-type theorems for disk-polygons in normed planes
A classical result of Dowker (Bull. Amer. Math. Soc. 50: 120-122, 1944)
states that for any plane convex body in the Euclidean plane, the areas of
the maximum (resp. minimum) area convex -gons inscribed (resp.
circumscribed) in is a concave (resp. convex) sequence. It is known that
this theorem remains true if we replace area by perimeter, the Euclidean plane
by an arbitrary normed plane, or convex -gons by disk--gons, obtained as
the intersection of closed Euclidean unit disks. The aim of our paper is to
investigate these problems for --gons, defined as intersections of
translates of the unit disk of a normed plane. In particular, we show that
Dowker's theorem remains true for the areas and the perimeters of circumscribed
--gons, and the perimeters of inscribed --gons. We also show that
in the family of origin-symmetric plane convex bodies, for a typical element
with respect to Hausdorff distance, Dowker's theorem for the areas of
inscribed --gons fails.Comment: 15 pages, 4 figure
Variance estimates for random disc-polygons in smooth convex discs
In this paper we prove asymptotic upper bounds on the variance of the number of vertices and missed area of inscribed random disc-polygons in smooth convex
discs whose boundary is C2+. We also consider a circumscribed variant of this probability model in which the convex disc is approximated by the intersection of random circles