Let $X$ be a nonsingular projective $n$-fold $(nge 2)$ of Fano or of general type with ample canonical bundle $K_X$ over an algebraic closed field $kappa$ of any characteristic. We produce a new method to give a bunch of inequalities in terms of all the Chern classes $c_1, c_2, cdots, c_n$ by pulling back Schubert classes in the Chow group of Grassmannian under the Gauss map. Moreover, we show that if the characteristic of $kappa$ is $0$, then the Chern ratios $(frac{c_{2,1^{n-2}}}{c_{1^n}}, frac{c_{2,2,1^{n-4}}}{c_{1^n}}, cdots, frac{c_{n}}{c_{1^n}})$ are contained in a convex polyhedron for all $X$. So we give an affirmative answer to a generalized open question, that whether the region described by the Chern ratios is bounded, posted by Hunt (cite{Hun}) to all dimensions. As a corollary, we can get that there exist constants $d_1$, $d_2$, $d_3$ and $d_4$ depending only on $n$ such that $d_1K_X^nlechi_{top}(X)le d_2 K_X^n$ and $d_3K_X^nlechi(X, mathscr{O}_X)le d_4 K_X^n$. If the characteristic of $kappa$ is positive, $K_X$ (or $-K_X$) is ample and $mathscr{O}_X(K_X)$ ($mathscr{O}_X(-K_X)$, respectively) is globally generated, then the same results hold.
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