Question:
What's the geometrical interpretation of Cauchy's inequality for $n = 2,3$?
When $n=2$:
CSI for $n=2$: $(a_1b_1+a_2b_2)^2 \leq (a_1^2+a_2^2)(b_1^2+b_2^2)$
Let $a_1=b_2=\sqrt{x}$ and $a_2=b_1=\sqrt{y}$: $(2\sqrt{xy})^2 \leq (x+y)^2 \implies \sqrt{xy} \leq \frac{x+y}{2}$
Geometrically, imagine the following: Let there be a semicircle with its diameter drawn. The diameter is divided into two (not necessarily equal) parts $x$ and $y$. From the division point, erect a perpendicular line segment intersecting the semicircle. This segment (which is a half-chord) is equal to $\sqrt{xy}$. From the center, erect a perpendicular line segment intersecting the semicircle. This segment (which is a radius or half-diameter) is equal to $\frac{x+y}{2}$. The CSI for $n=2$ claims that any chord of a circle is less than or equal to its diameter, which is true.
When $n=3$:
Any suggestions?
Best Answer
Once you have a geometric interpretation of Cauchy-Schwarz in the Cartesian plane, you have an interpretation in $n$-dimensional Cartesian space because:
Arbitrary non-proportional vectors $a$ and $b$ in $\mathbf{R}^{n}$ span a unique plane through the origin, and
This plane acquires the coordinate structure of the Cartesian plane as soon as you pick an orthonormal basis.
If you view Euclidean space in terms of coordinate-free length and angle, the task is even easier: An arbitrary plane in Euclidean $n$-space naturally acquires the structure of a Euclidean plane from the ambient concepts of length and angle.