I'm assuming you meant the most inexpensive method, rather than the least inexpensive. (Kind of like a double negative, sort of, almost.)
We're not talking quick, easy production of a truly scary sharp edge here, but Smokey Mountain Knife Works sells a little pocket sharpener, that has two "Vs" in it, one coarse and one fine. There are small ceramic rods in there. All one has to do is pull the knife blade thru them several times. Maybe many times if you're starting out from totally dull. I think it was a Gerber I got. Price was $2. I think Normark sells a single sided one for $2, and a double like the other for $3.
You're not going to be able to remove major quantities of steel in a hurry. If your blade(s) have a grind totally different than the "general" setup these sharpeners use, a lot of work may be necessary. I found a burr, or something on the point of one of my folders the other day. Was sure it would take a lot of work. Had just gotten the Gerber, so decided to check it out. It removed whatever was there in 5 passes thru the coarse side. I also touched up the blade on my old LST. Did about 6-7 passes thru coarse, then about as many thru fine. Definetly improved the edge noticeably.
I wear a beard, and am unduly fond of the hair on my arms, so don't use shaving checks on them, or my face. Besides, I have razors for trimming. Bics. But, the edge I put on the LST was at least as good as the original edge on it when I bought it 15 years ago or so.
Hope this helps -- particularly the tip about Bic disposables.
Woops! Brain fart here. Am not sure that incurring a $5 or so shipping charge for a $2 purchase is all that wise. In some mall kitchen supply stores, they sell the Firestone sharpener, the smaller one, for about $10. This has 2 revolving wheels of ceramic. I found a Fiskars sharpener with the same setup as the pocket sharpener I mentioned. It has a good handle which makes it easier to use. It cost $6, I think.
Forgot one thing. Use soap and water to clean the ceramic rods/wheels periodically. Dry them well. If you can -- I simply haven't tried it -- rotate the rods once in awhile too, so wear is even.
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Asi es la vida
Bugs
[This message has been edited by Bugs3x (edited 04-01-2001).]