MathDB

6

Part of 1987 Austrian-Polish Competition

Problems(1)

D(A) = \underset{d}{max} \underset{i}{min} \delta (P_i, d)

Source: Austrian Polish 1987 APMC

4/30/2020
Let CC be a unit circle and n≥1n \ge 1 be a fixed integer. For any set AA of nn points P1,...,PnP_1,..., P_n on CC define D(A)=maxd miniδ(Pi,d)D(A) = \underset{d}{max}\, \underset{i}{min}\delta (P_i, d), where dd goes over all diameters of CC and δ(P,ℓ)\delta (P, \ell) denotes the distance from point PP to line ℓ\ell. Let FnF_n be the family of all such sets AA. Determine Dn=minA∈FnD(A)D_n = \underset{A\in F_n}{min} D(A) and describe all sets AA with D(A)=DnD(A) = D_n.
combinatorial geometrycombinatoricsgeometrycircleSubsetsminmax