We control the probability of the uniform deviation between empirical and\ngeneralization performances of multi-category classifiers by an empirical L1\n-norm covering number when these performances are defined on the basis of the\ntruncated hinge loss function. The only assumption made on the functions\nimplemented by multi-category classifiers is that they are of bounded variation\n(BV). For such classifiers, we derive the sample size estimate sufficient for\nthe mentioned performances to be close with high probability. Particularly, we\nare interested in the dependency of this estimate on the number C of classes.\nTo this end, first, we upper bound the scale-sensitive version of the\nVC-dimension, the fat-shattering dimension of sets of BV functions defined on\nR^d which gives a O(1/epsilon^d ) as the scale epsilon goes to zero. Secondly,\nwe provide a sharper decomposition result for the fat-shattering dimension in\nterms of C, which for sets of BV functions gives an improvement from O(C^(d/2\n+1)) to O(Cln^2(C)). This improvement then propagates to the sample complexity\nestimate.\n