Nested cones and onion skins (Q1802418)

Let \(P\) be a Euclidean \(n\)-polytope with vertex set \(V\). For a fixed \(v_ 0 \in V\) denote \(V_ i:= \{v\in V: d(v_ 0,v) = i\}\) with the usual graph metric \(d\) on \(P\). Then the cones \(K_ i:= v_ 0 + \text{cone}(V_ i-v_ 0)\) spanned by \(v_ 0\) and \(V_ i\) are ordered by \(K_{i+1} \subseteq K\) for all \(i\in\mathbb{N}\). The author gives an elementary proof for this simple fact about ``nested cones'' and using polarity for a dual version about ``onion skins''.