MathDB

Let

S

be a nonempty closed bounded convex set in the plane. Let

K

be a line and

t

a positive number. Let

L_1

and

L_2

be support lines for

S

parallel to

K_1

, and let

\overline {L}

be the line parallel to

K

and midway between

L_1

and

L_2

. Let

B_S(K,t)

be the band of points whose distance from

\overline{L}

is at most

\left( \frac {t}{2} \right) w

, where

w

is the distance between

L_1

and

L_2

. What is the smallest

t

such that

S \cap \bigcap_K B_S (K, t) \ne \emptyset

for all

S

? (

K

runs over all lines in the plane.)

Problem Statement