Three old problems from the Polish school of mathematics

This note deals with three problems posed in the 1930s by two prominent members of the Polish school of mathematics. The first problem is known as the isometric Banach conjecture and deals with the simplest characterization of Hilbert spaces among Banach spaces. It was stated in 1932 by Stephan Banach in his book Théorie des opérations linéaires, 1987. The second is Problem 68 in the Scottish Book, 2015, posed by Stanislaw Ulam in 1936, and the third is the well known floating body problem stated by Stanislaw Ulam in 1935 and which is also known as Problem 19 of the Scottish Book.