By Jean-Marie Morvan

The reason of this ebook is to set the trendy foundations of the speculation of generalized curvature measures. This topic has a protracted heritage, starting with J. Steiner (1850), H. Weyl (1939), H. Federer (1959), P. Wintgen (1982), and maintains this present day with younger and really good mathematicians. within the final many years, a renewal of curiosity in arithmetic in addition to desktop technological know-how has arisen (finding new purposes in special effects, clinical imaging, computational geometry, visualization …).

Following a old and didactic strategy, the publication introduces the mathematical history of the topic, starting with curves and surfaces, occurring with convex subsets, soft submanifolds, subsets of confident achieve, polyhedra and triangulations, and finishing with floor reconstruction. We concentrate on the idea of ordinary cycle, which permits to compute and approximate curvature measures of a big type of soft or discrete gadgets of the Euclidean area. We supply particular computations while the item is a 2 or three dimensional polyhedron.

This e-book can function a textbook to any mathematician or laptop scientist, engineer or researcher who's attracted to the speculation of curvature measures.

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**Example text**

Definition 10. The Hausdorff distance d is defined as follows: for all elements P and Q of P(EN ), d(P, Q) = inf{r > 0 : P ⊂ Qr and Q ⊂ Pr }. 2 The Projection Map 49 Fig. , from l to m ). One has always |l − l | ≤ |mm | m’ l’ m l A One can easily check that d is effectively a distance on the set Pc (EN ) of compact subsets of EN . The main properties of dA are summarized in the following theorem. Theorem 10. Let A be a (nonempty) subset of EN . Then: 1. dA (m) = 0 ⇐⇒ m ∈ A. 2. The map dA : EN → R+ is uniformly Lipschitz and ∀m, m ∈ EN , dA (m) − dA (m ) ≤ d(m, m ).

1 Some authors simply use the term measure instead of signed measure. This can be done if no confusion is possible.

We shall prove that lim pn = p. n→∞ Since A is compact, there exists a subsequence of (pn )n∈N , which we still denote by (pn ), n ∈ N, which converges to a point q of A. Since the distance function dA is continuous, dA (mn ) tends to dA (m) = d(m, p). On the other hand, dA (mn ) = d(mn , A) = d(mn , pn ) tends to d(m, q). Hence, d(m, p) = d(m, q). Since m has a unique projection on A, p = q and lim pn = p. n→∞ This implies that all limiting values of the sequence (pn )n∈N equal p. Classically, we conclude that the sequence (pn )n∈N tends to p.