7. [30] Five points are chosen on a sphere of radius 1 . What is the maximum possible volume of their convex hull?
Answer: $\frac{\sqrt{3}}{2}$ Let the points be $A, B, C, X, Y$ so that $X$ and $Y$ are on opposite sides of the plane defined by triangle $A B C$. The volume is $1 / 3$ the product of the area of $A B C$ and sum of the distances from $X$ and $Y$ to the plane defined by $A B C$. The area of $A B C$ is maximized when the plane containing it intersects the sphere in the largest possible cross section, and $A B C$ is equilateral: this gives an area of $3 \sqrt{3} / 4$. Then, the sum of the distances from $X$ and $Y$ to the plane of $A B C$ is at most 2. This is clearly obtainable when $\mathrm{A}, \mathrm{B}$, and $\mathrm{C}$ form an equilateral triangle circumscribed by the equator of the sphere, and $\mathrm{X}$ and $\mathrm{Y}$ are at the poles, and we get a volume of $\sqrt{3} / 2$.