Small Ball Probabilities for Simple Random Tensors

We study the small ball probability of an order- simple random tensor X=X⁽¹⁾⊗⋯⊗ X⁽⁾ where X⁽ⁱ⁾, 1≤ i≤ are independent random vectors in Rⁿ that are log-concave or have independent coordinates with bounded densities. We show that the probability that the projection of X onto an m-dimensional subspace F falls within an Euclidean ball of length ε is upper bounded by ε/(-1)!(Clog(e/ε))^ and also this upper bound is sharp when m is small. We also established that a much better estimate holds true for a random subspace.