Part of the problem here is differing definitions of homeopathic.
Antioxidents are not homeopathic, they are nutrients found in food. Homeopathic was the concept that "like cures like", by putting in some sort of substance (often poisoness), and then diluting it out to a very small amount. Of course, the idea preceded the advent of modern chemistry and the understanding of molecules. When you dilute a solution, by removing 1/100 of it, and replacing the rest with water, and do so 30 times (as often the procedure), you reduce the molecules of the original substance by 1 times 10 to the minus 60! In other words, much much less then the number of molecules you began with . . .in other words, all of them.
Homepathic practiciners (sp) ignored this fact for nearly a hundred years. It is only recently that they have owned up to it and started creating some silly theories to get around it, such as that the water molecules themselves remember what substance used to be in it.
Antioxidents are found in food. They can be and are heavily studied through modern medical practice which means controlled studies. They certainly provide many benefits.