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For any $2\\ls q\\ls\\sqrt{n}$ and for $\\varepsilon \\in (\\varepsilon_0(q,n),1)$ we determine the inradius of a random $(1-\\varepsilon)n$-dimensional projection of $Z_q(\\mu)$ up to a constant depending polynomially on $\\varepsilon $. Using this fact we obtain estimates for the covering numbers $N(\\sqrt{\\smash[b]{q}}B_2^n,tZ_q(\\mu))$, $t\\gr 1$, thus showing that $Z_q(\\mu)$ is a $\\beta $-regular convex body. 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