An n -dimensional search problem with restricted questions

Abstract: The problem is the following: How many questions are necessary in the worst case to determine whether a pointX in then-dimensional Euclidean spaceR ⁿ belongs to then-dimensional unit cubeQ ⁿ, where we are allowed to ask which halfspaces of (n−1)-dimensional hyperplanes contain the pointX? It is known that ⌌3n/2⌍ questions are sufficient. We prove here thatcn questions are necessary, wherec≈1.2938 is the solution of the equationx log₂ x−(x−1) log₂ (x−1)=1.