@article{87ed753c57dd4deaac703c9f25f43243,

title = "A Water-Based Proof of the Cauchy–Schwarz Inequality",

author = "Mark Levi",

note = "Funding Information: The standard proof of the inequality of the title may appear a bit dry, and so I offer one that uses water. Consider n cylindrical cans with different cross-sectional areas ak that are connected by tubes as shown in the figure. With the valves closed, fill the kth can to an arbitrarily chosen height hk. Then open the valves, letting the water level out; basic mechanics tells us that the potential energy decreases:, with equality holding if and only if all the heights are equal at the outset. I claim that this amounts to the Cauchy–Schwarz inequality. What follows is a verification of this claim. The equalized level is given by, and the new, smaller potential energy is. So the statement amounts to(1) This is the Cauchy–Schwarz inequality in disguise. Indeed, driven by the desire to have on the right in (1), define by(2) Then multiplying the two definitions in (2) side-by-side (“sidewise”?) gives, or, so that (1) indeed amounts to the Cauchy–Schwarz inequality The above derivation is in the spirit of another derivation due to Tokieda [1] based on the energy dissipation. National Science Foundation;",

year = "2020",

month = jul,

day = "2",

doi = "10.1080/00029890.2020.1749514",

language = "English (US)",

volume = "127",

pages = "572",

journal = "American Mathematical Monthly",

issn = "0002-9890",

publisher = "Mathematical Association of America",

number = "6",

}