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Register a new userTotal variation distance estimates via $L^2$-norm for polynomials in log-concave random vectors
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Authors
Egor Kosov
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The paper provides an estimate of the total variation distance between distributions of polynomials defined on a space equipped with a logarithmically concave measure in terms of the $L^2$-distance between these polynomials.
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Let $C$ and $K$ be centrally symmetric convex bodies of volume $1$ in ${mathbb R}^n$. We provide upper bounds for the multi-integral expression begin{equation*}|{bf t}|_{C^s,K}=int_{C}cdotsint_{C}Big|sum_{j=1}^st_jx_jBig|_K,dx_1cdots dx_send{equation*} in the case where $C$ is isotropic. Our approach provides an alternative proof of the sharp lower bound, due to Gluskin and V. Milman, for this quantity. We also present some applications to randomized vector balancing problems.
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