MathDB

Anne has thought of a finite set

A \subseteq \mathbb{R}^2

. Bob does not know how many elements

A

has, but his goal is to completely determine

A

.
To achieve this, Bob can chooseany point

b \in \mathbb{R}^2

and ask Anne how far it is from

A

. Anne replies with the distance, defined as

min \{d(a, b) | a \in A\}

. (Here

d(a, b)

denotes the distance between points

a, b \in \mathbb{R}

.)
Bob can ask as many questions of this type as he wants, until he can determine A with certainty. a) Can Bob achieve his goal with finitely many questions? b) What if Anne tells Bob in advance that all points of A have both coordinates in the interval

\ [0, 1]\

? Note:

\mathbb{R}^2

is the set of points in the plane.

Problem Statement