Leave $ S ^ n $ Y $ B n + 1 $ be the sphere of the unit and the ball of the unit in $ mathbb {R} ^ {n + 1} $, and let $ M ^ n $ be a space of contractible dimension $ n $.

If necessary, suppose that $ M ^ n $ It is a contractable simplicial $ n $-complex.

Leave $ sigma colon S ^ n a M ^ n $ be a continuous map

Question 2 is a stronger variant of question 1.

## Question 1

Is there a continuous map? $ colon Bn + 1 a M ^ n $ such that for any $ b en B n + 1}, point $ b in mathbb {R} ^ {n + 1} $ is in the convex helmet of the preimage $ sigma -1 ((beta) b)?

## Question 2

Is there a continuous map? $ colon Bn + 1 a M ^ n $ and an involution $ f $ in $ S ^ n $ with $ sigma = sigma circ f $, so that for any $ b en B n + 1} is there any $ x in S ^ n $ such that $ sigma (x) = beta (b) $Y $ b $ is in the line segment between $ x $ Y $ f (x) $ in $ mathbb {R} ^ {n + 1} $?