On the structure of sets with positive reach

Rataj, Jan; Zaj́ıček, Luděk

doi:10.1002/mana.201600237

Cited by 26 publications

(41 citation statements)

References 31 publications

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“…Note that, in contrast with sets with positive reach which are defined by the geometrically illustrative “unique footpoint” property, there seems to be no purely geometric property characterizing WDC sets. Also, there is a number of results on the structure of sets of positive reach, see, e.g., , or the recent article . In the present article we prove some results on WDC sets, which are analogous to these results on sets of positive reach.…”

Section: Introductionsupporting

confidence: 67%

“…Remark [, Example 7.12(i)] shows that, in the above proposition, we cannot assert that

\partial M ∖ {(\partial M)}^{[d - 1]}

has zero

(d - 1)

‐dimensional Hausdorff measure even in the case when

M \subset R^{2}

is a set of positive reach.…”

Section: Results On the Structure Of Wdc Sets In Double-struckrdmentioning

confidence: 99%

“…Proof The case

h = 0

(hence, f is Lipschitz convex) was shown in [, Lemma 4.3, Proposition 4.4]. In the general case observe that

g_{+}^{'} false(x, v false) + g_{+}^{'} false(x, - v false) \geq 0

and

h_{+}^{'} false(x, v false) + h_{+}^{'} false(x, - v false) \geq 0

(

x \in Ω

v \in X

) since g and h are convex.…”

Section: Singular Sets Of Convex and DC Functionsmentioning

confidence: 89%

“…This section collects a few results on smallness of certain sets of singularities of convex and DC functions that will be needed in the sequel. We start with two propositions which are consequences of results from . Proposition Let X be a d ‐dimensional Hilbert space, let

1 \leq k < d

, let

Ω \subset X

be an open convex set, let g and h be Lipschitz convex functions on Ω, let

ε > 0

and denote

f ≔ g - h

.…”

Section: Singular Sets Of Convex and DC Functionsmentioning

confidence: 99%

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On the structure of WDC sets

Pokorný

Rataj

Zaj́ıček

2019

Mathematische Nachrichten

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WDC sets in ℝ were recently defined as sublevel sets of DC functions (differences of convex functions) at weakly regular values. They form a natural and substantial generalization of sets with positive reach and still admit the definition of curvature measures. Using results on singularities of convex functions, we obtain regularity results on the boundaries of WDC sets. In particular, the boundary of a compact WDC set can be covered by finitely many DC surfaces. More generally, we prove that any compact WDC set of topological dimension ≤ can be decomposed into the union of two sets, one of them being a -dimensional DC manifold open in , and the other can be covered by finitely many DC surfaces of dimension − 1. We also characterize locally WDC sets among closed Lipschitz domains and among lowerdimensional Lipschitz manifolds. Finally, we find a full characterization of locally WDC sets in the plane. K E Y W O R D S DC aura, DC domain, DC manifold, deformation retraction, Gauss-Bonnet formula, Lipschitz manifold, WDC set M S C ( 2 0 1 0 ) 26B25, 53C65 INTRODUCTIONFederer in his fundamental paper [10] unified the approaches of convex and differential geometry, introducing curvature measures for sets with positive reach and proving the kinematic formulas. Quite recently, curvature measures have been defined for a substantially larger class of so-called (locally) WDC sets [23], and the corresponding kinematic formulas have been proved [16]. The basic difference between the two named set classes is that, while sets with positive reach are closely related to semiconvex functions of several variables, WDC sets are related to DC functions (i.e., differences of two convex functions) instead. Following [14], we say that a locally Lipschitz function ∶ ℝ → [0, ∞) is an aura for a set ⊂ ℝ if = −1 {0} and 0 is a weakly regular value of (i.e., there exist no sequences → and → 0 such that ( ) > 0 = ( ) and ∈ ( ) are generalized gradients in the Clarke sense).This notion is motivated by the fact that has locally positive reach if and only if has a semiconvex aura [1]. By the definition, is WDC if and only if it has a DC aura. So each set with locally positive reach is a WDC set.Because of the theory built in [16,23], the following rough question naturally arises: What is the structure of a general WDC set? Note that, in contrast with sets with positive reach which are defined by the geometrically illustrative "unique footpoint" property, there seems to be no purely geometric property characterizing WDC sets. Also, there is a number of results on the structure of sets of positive reach, see, e.g., [10,21], or the recent article [24]. In the present article we prove some results on WDC sets, which are analogous to these results on sets of positive reach. 1596POKORNÝ ET AL. Boundaries of WDC setsWe show that the boundary of a compact WDC set in ℝ can be covered by finitely many DC hypersurfaces (i.e., graphs of Lipschitz DC functions of − 1 variables, see Proposition 6.1). Also, we show that a closed Lipschitz domain is locally WDC...

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Section: Introductionsupporting

confidence: 67%

“…Remark [, Example 7.12(i)] shows that, in the above proposition, we cannot assert that

\partial M ∖ {(\partial M)}^{[d - 1]}

has zero

(d - 1)

‐dimensional Hausdorff measure even in the case when

M \subset R^{2}

is a set of positive reach.…”

Section: Results On the Structure Of Wdc Sets In Double-struckrdmentioning

confidence: 99%

“…Proof The case

h = 0

(hence, f is Lipschitz convex) was shown in [, Lemma 4.3, Proposition 4.4]. In the general case observe that

g_{+}^{'} false(x, v false) + g_{+}^{'} false(x, - v false) \geq 0

and

h_{+}^{'} false(x, v false) + h_{+}^{'} false(x, - v false) \geq 0

(

x \in Ω

v \in X

) since g and h are convex.…”

Section: Singular Sets Of Convex and DC Functionsmentioning

confidence: 89%

1 \leq k < d

, let

Ω \subset X

be an open convex set, let g and h be Lipschitz convex functions on Ω, let

ε > 0

and denote

f ≔ g - h

.…”

Section: Singular Sets Of Convex and DC Functionsmentioning

confidence: 99%

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On the structure of WDC sets

Pokorný

Rataj

Zaj́ıček

2019

Mathematische Nachrichten

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“…Denoteε := . By [24,Corollary 20], the set C \ (a(S 1 )) ε has positive reach in the sense of Federer, and therefore, by [14,Remark 3.2], its boundary is rectifiable. In particular, it follows from Lemma 5.4 that there exists a net N with mesh sizeε whoseε blowup covers ∂S −ε d such that (5.8)…”

Section: Uniform Lower Bound On the Dominant Termmentioning

confidence: 99%

Spectrum of random perturbations of Toeplitz matrices with finite symbols

Basak

Paquette

Zeitouni

2020

Trans. Amer. Math. Soc.

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Consider an N × N Toeplitz matrix T N with symbol a(λ) := d 1 =−d 2 a λ , perturbed by an additive noise matrix N −γ E N , where the entries of E N are centered i.i.d. complex random variables of unit variance and γ > 1/2. It is known that the empirical measure of eigenvalues of the perturbed matrix converges weakly, as N → ∞, to the law of a(U ), where U is distributed uniformly on S 1 . In this paper, we consider the outliers, i.e. eigenvalues that are at a positive (N -independent) distance from a(S 1 ). We prove that there are no outliers outside spec T (a), the spectrum of the limiting Toeplitz operator, with probability approaching one, as N → ∞. In contrast, in spec T (a) \ a(S 1 ) the process of outliers converges to the point process described by the zero set of certain random analytic functions. The limiting random analytic functions can be expressed as linear combinations of the determinants of finite sub-matrices of an infinite dimensional matrix, whose entries are i.i.d. having the same law as that of E N . The coefficients in the linear combination depend on the roots of the polynomial Pz,a(λ) := (a(λ) − z)λ d 2 = 0 and semi-standard Young Tableaux with shapes determined by the number of roots of Pz,a(λ) = 0 that are greater than one in moduli. We remark that if one is interested in the case where d 1 = 0 but d 2 > 0, one may simply consider, when computing spectra, the transpose of T N or of M N = T N + ∆ N . For this reason, the restriction to d 1 > 0 does not reduce generality.

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A note on subsets of positive reach

Lytchak

2023

Mathematische Nachrichten

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On the structure of sets with positive reach

Cited by 26 publications

References 31 publications

On the structure of WDC sets

On the structure of WDC sets

Spectrum of random perturbations of Toeplitz matrices with finite symbols

A note on subsets of positive reach

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