Hölder-Łojasiewicz inequalities for volumes of tame objects (Q2231157)

Let \(\mathcal{D}\) be an \(o\)-minimal structure over the real field \(\mathbb{R}\) and \(K\subset \mathbb{R}^{m}\) a compact subset. A family \((S_{t})_{t\in T}\), \(T\subset \mathbb{R}^{p}\), of subsets of \(K\) is \textit{definiable family of subsets of }\(K\) if \(T\) is a definiable set (in \(\mathcal{D}\)) and there exists a definiable set \(S\subset T\times K\) such that \(S_{t}=\{x\in K:(t,x)\in S\}\). The author gives some uniform estimates of the volumes (in the Hausdorff measures \(\mathcal{H}^{k})\) of the images (and pre-images) of such definiable families through definiable maps. For instance, for images he proves: Let \(h:K\rightarrow \mathbb{R}^{n}\) be a continuous definable map. Let \( (S_{t})_{t\in T}\) be a definiable family of subsets of \(K\) with \(\dim S_{t}\leq k\) for all \(t\in T.\) Then there exists an odd, strictly increasing continuous definiable\ bijection \(\varphi \) from \(\mathbb{R}\) onto \(\mathbb{R }\) such that \[ \mathcal{H}^{k}(h(S_{t}))\leq \varphi (\operatorname{diam}(S_{t}))\ \ \ \text{for all }t\in T. \] He gives refined estimations under additional assumptions on the \(o\)-minimal structure \(\mathcal{D}.\) Similar results are also for preimages of such families.