Directional lower porosity (Q2510982)

Let \(X\) be a metric space. A set \(P\subset X\) is \textit{lower porous} at \(x\in P\) if there exists \(\rho>0\) and \(r_0>0\) such that for every \(0<r<r_0\) there exists \(y\in X\) with \(d(x,y)<r\) and \(B(y,\rho r)\cap P=\emptyset\). If \(X\) is a Banach space, then \(P\subset X\) is \textit{lower porous at \(x\in P\) in direction \(v\in X\)} if the points \(y\) in the definition above can be chosen on the line through \(x\) in direction \(v\). The author constructs a set in \(\mathbb{R}^2\) which is lower porous but not \(\sigma\)-directionally lower porous.