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

Source: Austrian Polish 1987 APMC

April 30, 2020

combinatorial geometrycombinatoricsgeometrycircleSubsetsminmax

Problem Statement

Let

C

be a unit circle and

n \ge 1

be a fixed integer. For any set

A

of

n

points

P_1,..., P_n

on

C

define

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

, where

A

goes over all diameters of

D_n = \underset{A\in F_n}{min} D(A)

and

\delta (P, \ell)

denotes the distance from point

P

to line

\ell

. Let

F_n

be the family of all such sets

A

. Determine

D_n = \underset{A\in F_n}{min} D(A)

and describe all sets

A

with

D(A) = D_n

.

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