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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?

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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.