# Lion and Man

Lion and Man

A man and a lion are inside a disk of radius 1 with center . Both have a top speed of 1. Can the man choose a strategy to avoid being captured by the lion?

S

To use the Demonstration, alternate the man's move (button) and the lion's move (choose a point inside the small circle and Alt+click).

A solution consists of a polygonal line of length , with , where . The lion can run a path of the same length and can choose the strategy so that the distance converges to 0, but only as time goes to infinity.

MM…MM…

1

2

k

k+1

s++⋯++⋯

1

2

1

3

1

k+1

s=1-s

1

s=M=MS

1

1

1

LL…LL…

1

2

k

k+1

ML

k

k

The man's strategy is as follows. The point is constructed toward the center of the disk so that the segment is of length and perpendicular to . The square of the distance of to the center of the disk is

M

k+1

MM

k

k+1

s

k+1

ML

k

k

M

k+1

S

MS≤MS+≤s+s++⋯+<s+s++⋯+=

k+1

2

k

2

s

k+1

2

1

2

2

1

2

2

1

3

2

1

(k+1)

2

1

2

2

1

1·2

1

2·3

1

k·(k+1)

s+s1-<s+s<(s+s)=1.

1

2

2

1

k+1

1

2

2

1

2

So the man is always inside the disk (explanation 1 for ).

k=1

The distance between the man and the lion is (explanation 2 for ).

ML≥->0

k+1

k+1

ML+

k

k

2

s

k+1

2

s

k+1

k=1