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A classical result of Dowker (Bull. Amer. Math. Soc. 50: 120-122, 1944)states that for any plane convex body $K$ in the Euclidean plane, the areas ofthe maximum (resp. minimum) area convex $n$-gons inscribed (resp.circumscribed) in $K$ is a concave (resp. convex) sequence. It is known thatthis theorem remains true if we replace area by perimeter, the Euclidean planeby an arbitrary normed plane, or convex $n$-gons by disk-$n$-gons, obtained asthe intersection of $n$ closed Euclidean unit disks. The aim of our paper is toinvestigate these problems for $C$-$n$-gons, defined as intersections of $n$translates of the unit disk $C$ of a normed plane. In particular, we show thatDowker's theorem remains true for the areas and the perimeters of circumscribed$C$-$n$-gons, and the perimeters of inscribed $C$-$n$-gons. We also show thatin the family of origin-symmetric plane convex bodies, for a typical element$C$ with respect to Hausdorff distance, Dowker's theorem for the areas ofinscribed $C$-$n$-gons fails.

Dowker-type theorems for disk-polygons in normed planes