Weighted covering numbers of convex sets (Q531815)

The ``classical'' covering number \(N(K,T)\) for \(K,T \subset \mathbb{R}^d\) is the minimal number \(N\) over all possible coverings \(\bigcup_{i=1}^N(x_i+T) \supset K\) of \(K\) by \(T\). The authors generalize this notion in the following way: A sequence of \(N\) pairs \((x_i,w_i) \in K \times \mathbb{R}^+\) is called a weighted covering of \(K\) by \(T\) if for all \(x \in K\) holds \(\sum{w_i \mathbf{1}_{T+x_i}(x)} \geq 1\) where \(\mathbf{1}_A\) denotes the indicator function of \(A\). The infimal total weight \(\sum_1^Nw_i\) over all weighted coverings of \(K\) by \(T\) is called the weighted covering number \(N_w(K,T)\) of \(K\) by \(T\). Analogously, the \(T\)-separation number \(M(K,T)\) in \(K\) is generalized to the weighted \(T\)-separation number \(M_w(K,T)\). The main results of the paper are as follows: For convex bodies \(K,T\) with symmetric \(T=-T\) holds \(N(K,2T)\leq N_w(K,T)\leq N(K,T)\) and \(N_w(K,T)=M_w(K,T)\).