Algebra and Geometry | Part IA, 2006

Prove the Cauchy-Schwarz inequality,

xyxy|\mathbf{x} \cdot \mathbf{y}| \leqslant|\mathbf{x}||\mathbf{y}|

for two vectors x,yRn\mathbf{x}, \mathbf{y} \in \mathbb{R}^{n}. Under what condition does equality hold?

Consider a pyramid in Rn\mathbb{R}^{n} with vertices at the origin OO and at e1,e2,,en\mathbf{e}_{1}, \mathbf{e}_{2}, \ldots, \mathbf{e}_{n}, where e1=(1,0,0,),e2=(0,1,0,)\mathbf{e}_{1}=(1,0,0, \ldots), \mathbf{e}_{2}=(0,1,0, \ldots), and so on. The "base" of the pyramid is the (n1)(n-1) dimensional object BB specified by (e1+e2++en)x=1,eix0\left(\mathbf{e}_{1}+\mathbf{e}_{2}+\cdots+\mathbf{e}_{n}\right) \cdot \mathbf{x}=1, \mathbf{e}_{i} \cdot \mathbf{x} \geqslant 0 for i=1,,ni=1, \ldots, n.

Find the point CC in BB equidistant from each vertex of BB and find the length of OC.(CO C .(C is the centroid of BB.)

Show, using the Cauchy-Schwarz inequality, that this is the closest point in BB to the origin OO.

Calculate the angle between OCO C and any edge of the pyramid connected to OO. What happens to this angle and to the length of OCO C as nn tends to infinity?

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