Consider the unit hypersphere (with radius r = 1). Inside the hypersphere inscribe a hypercube (i.e., the largest hypercube you can fit inside the hypersphere). An example in two dimensions is shown in Figure. Answer the following questions:  

(a) Derive an expression for the volume of the inscribed hypercube for any given dimensionality d. Derive the expression for one, two, and three dimensions, and then generalize to higher dimensions.

(b) What happens to the ratio of the volume of the inscribed hypercube to the volume of the enclosing hypersphere as d → ∞? Again, give the ratio in one, two and three dimensions, and then generalize.

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lets derive the expression for the volume of the inscribed hypercube in a unit hypersphere and analyze its ratio to the volume of the hypersphere as the dimensionality increases Volume of the Inscribe... View the full answer