Bounds of Hausdorff Measures V2(3-2009)

Bounds of Hausdorff Measures V2(3-2009)

BOUNDS OF HAUSDORFF MEASURES OF TAME SETS ˆ LOI AND PHAN PHIEN TA LE Abstract In this paper we present some bounds of Hausdorff measures of objects definable in o-minimal structures: sets, fibers of maps, inverse images of intervals of maps, etc Moreover, we also give some explicit bounds for semi-algebraic or semi-Pfaffian cases, which depend only on the combinatoric data representing the objects involved Considering the upper bounds for the lengths of curves contained in a disk, the areas of surfaces in a ball, or generally, the Hausdorff measures of subsets of a ball, one can see that if the numbers of points of the intersections of the curves or the surfaces with the generic lines are bounded, then their lengths or areas could be estimated Note that, spirals or oscillations not have finite numbers of points of intersections with generic lines, so they can have infinite lengths in certain disks The objects of o-minimal structures have the finiteness of number of connected components (see [D], [D-M], [C] and [L1]), and integral-geometric methods allow us to estimate Hausdorff measures of sets via the numbers of connected components of the intersections of the sets with generic affine subspaces of appropriate dimensions (see [F]) For these reasons, in this paper, we shall use integral-geometric methods to give some estimates of Hausdorff measures of objects definable in o-minimal structures: sets, fibers of maps, inverse images of intervals of maps, etc They can be considered as a generalization and refinement of some results of [H] Moreover, we also give some explicit bounds for semi-algebraic and semi-Pfaffian cases, which depend only on the combinatoric data representing the objects involved These relate to some results in [Y-C] and [D-K] In section we shall give some definitions The results and examples will be started and proved in sections - Definitions We give here some definitions and notations that will be used later 1.1 O-minimal structures An o-minimal structure on the real field (R, +, ·) is a sequence D = (Dn )n∈N such that the following conditions are satisfied for all n ∈ N: • Dn is a Boolean algebra of subsets of Rn 1991 Mathematics Subject Classification 32B20, 58C27, 14P10 Key words and phrases o-minimal structures, Hausdorff measures 39 40 ˆ LOI AND PHAN PHIEN TA LE • If A ∈ Dn , then A × R and R × A ∈ Dn+1 • If A ∈ Dn+1 , then π(A) ∈ Dn , where π : Rn+1 → Rn is the projection on the first n coordinates • Dn contains {x ∈ Rn : P (x) = 0}, for every polynomial P ∈ R[X1 , , Xn ] • Each set in D1 is a finite union of intervals and points A set belonging to D is said to be definable (in that structure) Definable maps in structure D are maps whose graphs are definable sets in D The class of semi-algebraic sets and the class of sub-Pfaffian sets ([K] and [W]) are examples of such structures, and there are many interesting classes of sets which have been proved to be o-minimal For important properties of ominimal structures we refer the readers to [D], [D-M], [C], [L1] and [W] Note that by Cell Decomposition [D] Ch.3 Th.2.11, the dimension of a definable set A is defined by dim A = max{dim C : C is a C submanifold contained in A} In this note we fix an o-minimal structure on (R, +, ·) “Definable” means definable in this structure 1.2 Diagrams of semi-algebraic sets Let A ⊂ Rm be a semi-algebraic set i represented by A = ∪pi=1 ∩jj=1 Aij , where each Aij has the form: {x ∈ Rm : pij (x) ? 0}, where pij is a polynomial of degree dij and ? ∈ {>, ≥} The set of data D = D(A) = (m, p, j1 , , jp , (dij )i=1, ,p; diagram of the set A j=1, ,ji ) is called the 1.3 Formats of Semi-Pfaffian sets Pfaffian chains A Pfaffian chain of length r ≥ and degree α ≥ in an open domain U ⊆ Rm is a sequence of analytic functions f = (f1 , , fl ) in U satisfying a system of Pfaffian equations ∂ fi (x) = Pij (x, f1 (x), , fi (x)), ∀x ∈ U (1 ≤ i ≤ l, ≤ j ≤ n) ∂ xj where Pij are polynomials of degree not exceeding α Pfaffian functions We say that q is a Pfaffian function of degree β with the Pfaffian chain f if there exists a polynomial Q of degree not exceeding β such that q(x) = Q(x, f1 (x), , fl (x)), ∀x ∈ U QF formulae Let P = {p1 , , ps } be a set of Pfaffian functions A quantifier-free formula (QF formula) with atoms in P is constructed as follows: • An atom is of the form pi ? 0, where ≤ i ≤ s and ? ∈ {=, ≤, ≥} It is a QF formula; BOUNDS OF HAUSDORFF MEASURES OF TAME SETS 41 • If Φ and Ψ are QF formulae, then their conjunction Φ ∧ Ψ, their disjunction Φ ∨ Ψ, and the negation ¬Φ are QF formulae Semi-Pfaffian sets A set A ⊆ U is called semi-Pfaffian if there exists a finite set P of Pfaffian functions and a QF formula Φ with atoms in P such that A = {x ∈ U : Φ(x)} Formats of semi-Pfaffian sets Let A be a semi-Pfaffian set as above Then the format of A is the set of data F = F (A) = (m, l, α, β, s), where m is the number of variables, l is the length of f , α is the maximum of the degrees of the polynomials Pij , β is the maximum of the degrees of the functions in P,and s is the number of the functions in P 1.4 A formula of integral geometric measure Let m be a positive integer For each k ∈ {0, , m}, let Hk (A) denote the k-dimensional Hausdorff measure of A ⊂ Rm Let O∗ (m, k) denote the space of all orthogonal projections of Rm onto Rk , i.e O∗ (m, k) = {p| p : Rm → Rn linear and p ◦ p∗ = id Rk } The orthogonal group O(m) acts transitively on O∗ (m, k) through right mul∗ tiplication This action induces a unique invariant measure θm,k over O∗ (m, k) ∗ with θm,k [O∗ (m, k)] = The Cauchy-Crofton formula By [F] 2.10.15 and 3.2.26, for every Borel subset B of Rm , we have Z Z k ∗ H (B) = c(m, k) #(B ∩ p−1 (y))dydθm,k p O∗ (m,k) Rk Γ( m+1 )Γ( 12 ) where c(m, k) = k+1 m−k+1 , and Γ(s) = Γ( )Γ( ) Z +∞ e−t ts−1 dt (s > 0) Uniform bounds of the Betti numbers of the fibers Proposition Let f : A → Rn be a continuous definable map Let i ∈ N Then there exists a positive number Mi , such that the i-th Betti numbers of the fibers of f are bounded by Mi Bi (f −1 (y)) ≤ Mi , for all y ∈ Rn In particular, the numbers of connected components of the fibers of f are uniformly bounded Moreover, if f is semi-algebraic (reps semi-Pfaffian), then Mi only depends on the diagram (resp the format) of f 42 ˆ LOI AND PHAN PHIEN TA LE Proof By Hardt’s Trivialization Theorem [D] Ch.9 Th.1.2, there is a finite partition f (A) = C1 ∪ ∪ CM of A into definable sets Ci such that f is definable trivial over each Ci Hence the family of the fibers of f has only finitely many embedded definable topological types So the Betti numbers are uniformly bounded Moreover, when f is semi-algebraic or semi-Pfaffian, by [B],[G-V] or [K],[Z],[G-V-Z], the Betti numbers are bounded by constants depending only on the diagram or the format of f  Hausdorff measures of definable sets Let A be a subset of Rm For each k ∈ {0, , m}, define B0,m−k (A) = sup{B0 (A ∩ p−1 (y)) : p ∈ O∗ (m, k), y ∈ Rk } Note that if A is definable, then, applying Proposition to the canonical projection {(x, p, y) ∈ A × O∗ (m, k) × Rk : p(x) = y} → {(p, y) ∈ O∗ (m, k) × Rk } we get the boundedness of B0,m−k (A) Moreover, if A is semi-algebraic or semi-Pfaffian, then B0,m−k (A) is bounded by an explicit constant depending only on the diagram or the format of A (see the examples below) Theorem Let A, B be definable subsets of Rm Suppose B is compact, dim A = k, and A ⊂ B Then Hk (A) ≤ c(m, k)B0,m−k (A) Hk (p(B)) sup p∈O∗ (m,k) If moreover A, B are semi-algebraic or semi-Pfaffian sets, then Hk (A) ≤ C Hk (p(B)) sup p∈O∗ (m,k) where C is a constant depending only on the diagram or the format of A Proof By [D] Ch.4 Prop.1.5, for each p ∈ O∗ (m, k), dim(B ∩ p−1 (w)) ≤ 0, and dim(A ∩ p−1 (w)) ≤ 0, for all w ∈ Rk outside a definable set of dimension less than k By the Cauchy-Crofton formula, we get the estimate Z Z k ∗ H (A) = c(m, k) #(A ∩ p−1 (w))dwdθm,k p O∗ (m,k) Rk Z Z ∗ ≤ c(m, k)B0,m−k (A) 1p(A) dwdθm,k p ∗ k O (m,k) R Z Z ∗ ≤ c(m, k)B0,m−k (A) 1p(B) dwdθm,k p O∗ (m,k) ≤ c(m, k)B0,m−k (A) sup Rk Hk (p(B)) p∈O∗ (m,k) The last assertion is followed by Proposition  BOUNDS OF HAUSDORFF MEASURES OF TAME SETS 43 Corollary (c.f [Y-C] and [D-K]) Let A be a definable subset of Rm of dimension k Then for any ball Brm of radius r in Rm , Hk (A ∩ Brm ) ≤ c(m, k)B0,m−k (A)Vol k (B1k )rk Proof From the preceding theorem, we get Hk (A ∩ Brm ) ≤ c(m, k)B0,m−k (A)Hk (Brk ) = c(m, k)B0,m−k (A)Vol k (B1k )rk  Example Algebraic case When A ⊂ R˜m is a k-dimensional algebraic set of degree d, then Hk (A ∩ Brm ) ≤ c(m, k)dVol k (B1k )rk In particular, when A is an algebraic curve of degree d in the plane, then the length l(A ∩ Br2 ) ≤ c(2, 1)d2r = πdr Semi-algebraic case Generally, when A ⊂ Rm is a k-dimensional semialgebraic set of diagram D = (m, p, j1 , , jp , (dij )i=1, ,p;j=1, ,ji ), then Hk (A ∩ Brm ) ≤ c(m, k)B0 (D)Volk (B1k )rk where B0 (D) = p X di (di − 1) i=1 m−1 , with di = ji X dij (see [Y-C], [B]) j=1 Semi-Pfaffian case We say that U is a domain of bounded complexity γ for the Pfaffian chain f = (f1 , , fl ) if there exists a function g of degree γ in the chain f such that the sets {g ≥ ε} form an exhausting family of compact subsets of U for ε  We call g an exhausting function for U Let A be a k-dimensional semi-Pfaffian set defined by a fixed Pfaffian chain f = (f1 , , fl ) of degree α in a domain U ⊆ Rm with format (m, l, α, β, s), where U is a domain of bounded complexity γ for f Using [Z] Remark.1.30, Th.2.25, Remark 2.26, and applying Corollary 1, we get Hk (A ∩ Brm ) ≤ c(m, k)(4s + 1)d V(m, l, α, β ∗ , γ)Vol k (B1k )rk where V(m, l, α, β ∗ , γ) = l(l−1) γ β ∗ (α + β ∗ − 1)n−1 [n(α + β ∗ − 1) + γ + min(m, l)α]l with β ∗ = max(β, γ) Uniform bounds of Hausdorff measures of definable fibers Let f : A → Rn be a definable map, where A ⊂ Rm For each k ∈ {0, , dim A}, let Ik (f ) = {y ∈ Rn : dim f −1 (y) ≤ k} Then, by [D] Ch.4.1.6, Ik (f ) is definable Let B0,m−k (f ) = sup{B0 (f −1 (y) ∩ p−1 (w) ∩ B m (a, r)) : y ∈ Ik (f ), p ∈ O∗ (m, k), w ∈ Rk , a ∈ Rm , r > 0} ˆ LOI AND PHAN PHIEN TA LE 44 Note that applying Proposition to the canonical projection {(x, y, p, w, a, r) ∈ Rm × Rn × O∗ (m, k) × Rk × Rm × R : x ∈ A, y ∈ Ik (f ), f (x) = y, p(x) = w, kx − ak ≤ r} → {(y, p, w, a, r) ∈ Rn × O∗ (m, k) × Rk × Rm × R} we have the boundedness of B0,m−k (f ) When f is semi-algebraic (resp semiPfaffian), then B0,m−k (f ) is bounded by a constant depending only on the diagram (resp format) of f Theorem Let f : A → Rn be a continuous definable map, where A is a compact suset of Rm Then for each k ∈ {0, , dim A}, we have Hk (f −1 (y)) ≤ c(m, k)B0,m−k (f ) Hk (p(A)), for all y ∈ Ik (f ) sup p∈O∗ (m,k) In particular, if f is semi-algebraic or semi-Pfaffian map, then Hk (f −1 (y)) ≤ Ck sup Hk (p(A)), for all y ∈ Ik (f ) p∈O∗ (m,k) where Ck is a constant depending only on the diagram or the format of f Proof By [D] Ch.4, Prop.1.6, for each p ∈ O∗ (m, k) and y ∈ Ik (f ), dim(f −1 (y)∩ k p−1 λ (w)) ≤ 0, for all w ∈ R outside a definable set of dimension less than k By the Cauchy-Crofton formula, when y ∈ Ik (f ), we get Z Z k −1 ∗ H (f (y)) = c(m, k) #(f −1 (y) ∩ p−1 (w))dwdθm,k p O∗ (m,k) Rk Z Z ∗ ≤ c(m, k)B0,m−k (f ) 1p(A) dwdθm,k p O∗ (m,k) ≤ c(m, k)B0,m−k (f ) sup Rk Hk (p(A)) p∈O∗ (m,k) If f is semi-algebraic or semi-Pfaffian, then using the note above we have the last assertion  Corollary Let f : A → Rn be a continuous definable map, where A ⊂ Rm Then for each k ∈ {0, , dim A} and for any ball Brm of radius r in Rm , Hk (f −1 (y) ∩ Brm ) ≤ c(m, k)B0,m−k (f )Vol k (B1k )rk , for all y ∈ Ik (f ) In particular, if f is semi-algebraic or semi-Pfaffian map, then Hk (f −1 (y) ∩ Brm ) ≤ Ck rk , for all y ∈ Ik (f ) where Ck is a constant depending only on the diagram or the format of f Example The family ofX algebraic sets Let A = {(x, a) : x = (x1 , , xm ) ∈ Rm , a = (aα )α∈Nm ,|α|≤d , aα xα = 0}, and f be the projection (x, a) 7→ a Then α Aa = A ∩ f −1 (a) be algebraic subsets of Rm of degree ≤ d Applying the BOUNDS OF HAUSDORFF MEASURES OF TAME SETS 45 theorem, one gets the estimates from above, similar to Example 1, for the k-dimensional Hausdorff measures of algebraic sets containing in the balls of radius r Hk (f −1 (a) ∩ Brm ) = Hk (Aa ∩ Brm ) ≤ c(m, k)dVol k (B1k )rk , when a ∈ Ik (f ) Fewnomial case Let α1 , , αq ∈ Nm Consider the family of algebraic surfaces in the positive orthant determined by the ‘fewnomials’ having only at most the monomials xαi , i = 1, , q: A = {(x, a) : x = (x1 , , xm ) ∈ Rm , a = (a1 , , aq ) ∈ Rq , x1 > 0, , xm > 0, q X xαi = 0} i=1 Let f be the projection (x, a) 7→ a and Aa = A ∩ f −1 (a) When k = m − 1, and dim Aa ≤ m − from the theorem we have the following estimates: Estimate Since Aa is a semi-algebraic set of diagram (m, 1, m+1, (1, , 1, d)), with d = maxi |αi |, using the Thom-Milnor bound (see [Y-C]), we get Hm−1 (Aa ∩ Brm ) ≤ c(m, m − 1)B0 (D(Aa ))Volm−1 (B1m−1 )rm−1 where B0 (D(Aa )) = 21 (m + d)(m + d − 1)m−1 Estimate Using [K] Ch.III Corol.5, we get Hm−1 (Aa ∩ Brm ) ≤ c(m, m − 1)B0 (f )Volm−1 (B1m−1 )rm−1 , q(q−1) where B0 (f ) = 2 (2m)m−1 (2m2 − m + 1)q Real exponential case Let A = {(x, α1 , , αq , a1 , , aq ) : x = (x1 , , xm ) ∈ Rm , α1 , , αq ∈ Rm , a1 , , aq ∈ R, x1 > 0, xm > 0, q X xαi = 0} i=1 and f be the projection (x, α1 , , αq , a1 , , aq ) 7→ (α1 , , αq , a1 , , aq ) Then f is definable in the structure Rexp (see [D-M]) Applying the theorem, one can get the estimates from above for the k-dimensional Hausdorff measures of sets defined by real exponential polynomials having at most q monomials, that contained in the intersection of the positive orthant and the balls of radius r Let Φ1 denote the set of all odd, strictly increasing C definable bijection from R to R and flat at Theorem Let f : A → Rn be a continuous definable map, and A ⊂ Rm be a compact set Then for each k ∈ {0, , dim A}, there exists ϕ ∈ Φ1 , such that Hk+1 (f −1 ([y, z])) ≤ ϕ−1 (ky − zk), whenever [y, z] ⊂ Ik (f ) ˆ LOI AND PHAN PHIEN TA LE 46 In particular, if f is semi-algebraic, then Hk+1 (f −1 ([y, z])) ≤ Cky − zkα , whenever [y, z] ⊂ Ik (f ) where α and C are positive constants depending only on the diagram of f Proof The proof is an adaptation of that of [H] Th.5 For k = : Since I0 (f ) is compact and the fibers of f over I0 (f ) are finite, by Trivialization [D] Ch.9, f −1 (I0 (f )) = ∪Jj=1 Aj , where Aj is a compact definable set and f |Aj is injective For each j ∈ {1, , J}, applying generalized Lojasiewicz inequalitiy and Hăolder continuity [D-M] 4.20-21 to (f |Aj )−1 , we get ϕj,1 , ϕj,2 ∈ Φ1 , such that −1 −1 H1 ((f |Aj )−1 ([y, z])) ≤ ϕ−1 j,1 (k(f |Aj ) (y) − (f |Aj ) (z)k) −1 −1 ≤ ϕ−1 j,1 ◦ ϕj,2 (ky − zk) = ϕj (ky − zk) Therefore, there exists ϕ ∈ Φ1 , such that H (f −1 ([y, z])) ≤ J X −1 ϕ−1 j (ky − zk) ≤ ϕ (ky − zk) j=1 For k ≥ 1: let Gk (Rm ) denote the Grassmannian of k-dimensional linear subspaces of Rm Define dist(L, L0 ) = sup{d(x, L0 ) : x ∈ L, kxk = 1}, for L ∈ Gk (Rm ), L0 ∈ Gl (Rm ) Let π : Rm → Rk denote the canonical projection Choose a finite subset I of O(m) and δ > 0, so that for each L ∈ Gk (Rm ), there exists g ∈ I so that dist(L, (π ◦ g)−1 (0)) > δ By [L2] we can choose a stratification S of A satisfying Whitney’s condition (a), so that for each S ∈ S, rankf |S is constant and either f (S) ⊂ Ik (f ) or f (S) ∩ Ik (f ) = ∅ Let J = {S ∈ S : dim S − rank f |S = k} We can refine the stratification so that for each g ∈ I and T ∈ J , the definable function d(T, g)(x) = dist(Tx T ∩ f −1 (f (x)), (π ◦ g)−1 (0)) − δ has constant sign on T For each S ∈ S\J we have dim(S ∩f −1 (y)) ≤ k −1 for all y ∈ Ik (f ), therefore, Hk+1 (f −1 ([y, z])\ ∪T ∈J T ) = whenever [y, z] ⊂ Ik (f ) For each T ∈ J , there is a gT ∈ I so that d(T, gT ) is positive on T Hence, by Whitney’s condition (a), dim(f −1 (y) ∩ (π ◦ gT )−1 (w) ∩ cl(T )) ≤ 0, for all y ∈ Ik (f ), w ∈ Rk For each g ∈ I, let Ag = ∪{cl(T ) : T ∈ J , gT = g} Using the coarea formula [F] Th.3.2.22 (3) and applying case k = with A := Ag and f := (f, pλ ◦ g)|Ag , BOUNDS OF HAUSDORFF MEASURES OF TAME SETS 47 we get Hk+1 (f −1 ([y, z])) ≤ X Hk+1 (g(Ag ∩ f −1 ([y, z])) g∈I = XZ g∈I = Jk π dHk+1 g(Ag XZ ∩f −1 ([y,z])) H1 (g(Ag ∩ f −1 ([y, z])) ∩ π −1 (w))dw g∈I = XZ H1 (Ag ∩ f −1 ([y, z]) ∩ g −1 (π −1 (w)))dw g∈I = XZ H1 (Ag ∩ (f, π ◦ g)−1 [(y, w), (z, w)])dw g∈I ≤ XZ 1π◦g(A) ϕ−1 (ky − zk)dw g∈I ≤ X Hk (π ◦ g(A))ϕ−1 (ky − zk) g∈I ≤ ϕ¯−1 (ky − zk) , where ϕ¯ ∈ Φ1 , ϕ¯ ≤ const.ϕ If f is a semi-algebraic map, then by the Lojasiewicz inequality ϕ has the form ϕ−1 (y) = Ckykα Moreover, by [B-R] Remark 2.3.13, C and α can be effectively bounded by the diagram of f The last assertion follows  Note that the above estimate is ‘effective’ not explicit Example a) In general, for semi-algebraic case one can not choose α = in the estimate of the preceding theorem, e.g for f (x) = xn with n ≥ 2, there does not exist √ any C > such that the lenght l(f −1 ([0, y]) = n y ≤ C|y|, for every y ∈ [0, 1] b) Let f (x) = e− |x| Then f is definable in the o-minimal structure Rexp , and 1 ] Since α → ∞, when y → 0, there does not f −1 ([0, y]) = [0, − ln |y| y ln |y| exist C, α > so that l(f −1 ([0, y]) ≤ C|y|α for all y ∈ [0, 1] Morse-Sard’s Theorem Theorem Let f : A → Rn be a definable map Suppose A = ∪i∈I Ci is a finite union of C definable manifolds Ci , such that f |Ci is of class C For each s ∈ N and i ∈ I, let [ Σs (f, Ci ) = {x ∈ Ci : rank df |Ci (x) < s} and Σs (f, A) = Σs (f, Ci ) i∈I 48 ˆ LOI AND PHAN PHIEN TA LE Then Cs (f, A) = f (Σs (f, A)) is a definable set of dimension < s In particular, Hs (Cs (f, A)) = Proof The proof is similar to [L3] S It is easy to see that Σs (f, Ci ) is definable for each i ∈ I So Cs (f, A) = f ( i∈I Σs (f, Ci )) is definable Suppose, contrary to the assertion, that dim Cs (f ) ≥ s Then, by the Definable Choice [D-M] Th.4.5, there exist i ∈ I, a definable subset U of Cs (f, A) and a definable C mapping s : U → Σs (f, Ci ) such that f ◦ s = idU So rank df |Ci (s(y))ds(y) ≥ s, for all y ∈ U Hence, rank df |Ci (x) ≥ s, for all x ∈ s(U ) This is a contradiction  Note that, if we consider the class of all C p mappings f : Rm → Rn , then Morse-Sard’s Theorem requires the differentiability class of f H Whitney constructed in [W] an example of a C function f : R2 → R not constant on a connected set of critical points and hence H1 (C0 (f, R2 )) 6= Theorems 3.4.3 and 3.4.4 in [F] proved that: Given integers p ≥ 1, and ≤ s < m, the least number α such that Hα (Cs (f, Rm )) = 0, for every function f of class C p m−s mapping an open subset of Rm into some normed vector space Y , is s+ k Remarks The results in this paper still hold true for tame sets (see [D-M], [S], [T] for the definitions) with global changing to local Applying theorems and 2, one can get the explicit estimates for sub-Pfaffian case (see [G-V-Z]) References [B] S.Basu On bounds the Betti numbers and computing the Euler characterstics of semialgebraic sets, Discrete and Comput 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H.Whitney, A Function not Constant on Connected Set of Critical Points, Duke Math.J 1, 1935, 514-517 [Y-C] Y.Yomdin and G.Comte, Tame Geometry with Application in Smooth Analysis, LNM vol 1834, 2004 [Z] T.Zell, Quantitative study of semi-Pfaffian sets, School of Mathematics, Georgia Institue of Technology, 2003 department of mathematics, university of dalat, dalat, vietnam nhatrang pedagogical college, nhatrang, vietnam ... subsets of Rm of degree ≤ d Applying the BOUNDS OF HAUSDORFF MEASURES OF TAME SETS 45 theorem, one gets the estimates from above, similar to Example 1, for the k-dimensional Hausdorff measures of. .. followed by Proposition  BOUNDS OF HAUSDORFF MEASURES OF TAME SETS 43 Corollary (c.f [Y-C] and [D-K]) Let A be a definable subset of Rm of dimension k Then for any ball Brm of radius r in Rm , Hk... the number of variables, l is the length of f , α is the maximum of the degrees of the polynomials Pij , β is the maximum of the degrees of the functions in P,and s is the number of the functions