Lipschitz constants for the width and diameter functions of a convex body in R(n )are found in terms of its diameter and thickness (maximum and minimum of both functions). Also, a dual approach to thickness is proposed.

Given an inner function e, we study the quotient algebra H infinity/eH infinity, and more precisely, the norms of the inverses of its elements f. P. Gorkin, R. Mortini, and N. Nikolski introduced the property of e that a positive lower bound for |f | on the zero set of e implies invertibility of f and called it the Weak Embedding Property. When the fact that lower bounds come close to 1 implies that the inverse f-1 exists with norm close to 1, we say that e has the Sharp Invertibility Property (SIP). We prove that the SIP is equivalent to the maximal asymptotic growth of e away from its zero set. We also relate the SIP to the geometric properties of the sup-and sublevel sets of |e|, and to the Frostman shifts of e being Carleson-Newman Blaschke products. We finally study divisors of inner functions satisfying the SIP. We describe geometrically the zero set of inner functions such that all its divisors satisfy the SIP. We also characterize the closed subsets E of the unit circle of Lebesgue measure 0 such that any singular inner function associated to a singular measure supported on E is a divisor of an inner function satisfying the SIP. (c) 2026 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

We give precise estimates of some holomorphically invariant infinitesimal metrics near a pseudoconcave points in a wide family of “model” domains for that situation in ℂ^2. This extends to metrics (rather distances) the authors' previous results from arXiv:2503.19754 and also takes into account defining functions more general than just power functions.

We show that for bounded domains in ℂ^n with 𝒞^1,1 smooth boundary, if there is a closed set F of 2n-1-Lebesgue measure 0 such that ∂ Ω∖ F is 𝒞^2-smooth and locally pseudoconvex at every point, then Ω is globally pseudoconvex. Unlike in the globally 𝒞^2-smooth case, the condition “F of (relative) empty interior” is not enough to obtain such a result. We also give some results under peak-set type hypotheses, which in particular provide a new proof of an old result of Grauert and Remmert about removable sets for pseudoconvexity under minimal hypotheses of boundary regularity.

It is shown that the optimal upper and lower bounds for the Kobayashi distance near 𝒞^2,α-smooth strongly pseudoconvex boundary points obtained in L. Kosinski, N. Nikolov, A.Y. Okten: "Precise estimates of invariant distances on strongly pseudoconvex domains", Adv. Math. 478 (2025), 110388, remain true in the general 𝒞^2 strongly pseudoconvex setting. In fact, the upper bound is extended to the general 𝒞^1,1-smooth case. We also give upper and lower bounds for the Kobayashi distance near non-semipositive boundary points.

The (unbounded version of the) Lempert function l_D on a domain D⊂ℂ^d does not usually satisfy the triangle inequality, but on bounded 𝒞^2 -smooth strictly pseudoconvex domains, it satisfies a quasi-triangle inequality: l_D(a,c)≤ C( l_D(a,b)+l_D(b,c)) . We show that pseudoconvexity is necessary for this property as soon as D has a 𝒞^1 -smooth boundary. We also give estimates of the Lempert function and of other invariants in some domains which are models for local situations, and derive some general local bounds depending on the regularity of the boundary of a domain.

We study the relationship between the smoothness of a plane curve and that of its evolute, especially in the cases where the parent curve is no more two or three times continuously differentiable, and exhibit the same kind of apparent improvement in regularity: in the generic local situation, the evolute has one order of regularity less than the parent curve.

We study the gain in regularity of the distance to the boundary of a domain in ℝ^m . In particular, we show that if the signed distance function happens to be merely differentiable in a neighborhood of a boundary point, it and the boundary have to be 𝒞^1,1 regular. Conversely, we study the regularity of the distance function under regularity hypotheses of the boundary. Along the way, we point out that any solution to the eikonal equation, differentiable everywhere in a domain of the Euclidean space, admits a gradient which is locally Lipschitz.

We introduce the notion of locally visible and locally Gromov hyperbolic domains in C d \mathbb {C}^d . We prove that a bounded domain in C d \mathbb {C}^d is locally visible and locally Gromov hyperbolic if and only if it is (globally) visible and Gromov hyperbolic with respect to the Kobayashi distance. This allows to detect, from local information near the boundary, those domains which are Gromov hyperbolic and for which biholomorphisms extend continuously up to the boundary.

We prove that in a strongly pseudoconvex domain with smooth boundary, then the length of a geodesic for the Kobayashi-Royden infinitesimal metric between two points is bounded by a constant multiple of the Euclidean distance between the points.

In this note, we introduce the notion of visible boundary points with respect to Kobayashi distance for domains in complex euclidean space. Following the work of Sarkar, we obtain additive and multiplicative localization results about Kobayashi distance near visible boundary points. Then using the additive localization result, we show that visibility property with respect to Kobayashi distance is a local property of the boundary points and it does not depend on the domain.

We show that a domain in C n with C 2-smooth boundary which satisfies the visibility property is pseudoconvex. (c) 2024 Elsevier Masson SAS. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

AbstractLet $\mathcal {N}$ be the Nevanlinna class, and let B be a Blaschke product. It is shown that the natural invertibility criterion in the quotient algebra $\mathcal {N} / B \mathcal {N}$ , that is, $|f| \ge e^{-H} $ on the set $B^{-1}\{0\}$ for some positive harmonic function H, holds if and only if the function $- \log |B|$ has a harmonic majorant on the set $\{z\in \mathbb {D}:\rho (z,\Lambda )\geq e^{-H(z)}\}$ , at least for large enough functions H. We also study the corresponding class of positive harmonic functions H on the unit disc such that the latter condition holds. We also discuss the analogous invertibility problem in quotients of the Smirnov class.

We obtain explicit bounds on the difference between "local" and "global" Kobayashi distances in a domain of C-n as the points go toward a boundary point with appropriate geometric properties. We use this for the global comparison of various invariant distances. We provide some sharp estimates in dimension 1.

Let $$D\subset {\mathbb {C}}^n$$ be a bounded domain. A pair of distinct boundary points $$\{p,q\}$$ of D has the visibility property provided there exist a compact subset $$K_{p,q}\subset D$$ and open neighborhoods $$U_p$$ of p and $$U_q$$ of q, such that the real geodesics for the Kobayashi metric of D which join points in $$U_p$$ and $$U_q$$ intersect $$K_{p,q}$$ . Every Gromov hyperbolic convex domain enjoys the visibility property for any couple of boundary points. The Goldilocks domains introduced by Bharali and Zimmer and the log-type domains of Liu and Wang also enjoy the visibility property. In this paper we relate the growth of the Kobayashi distance near the boundary with visibility and provide new families of convex domains where that property holds. We use the same methods to provide refinements of localization results for the Kobayashi distance, and give a localized sufficient condition for visibility. We also exploit visibility to study the boundary behavior of biholomorphic maps.

It is shown that even a weak multidimensional Suita conjecture fails for any bounded non-pseudoconvex domain with C1 boundary: the product of the Bergman kernel by the volume of the indicatrix of the Azukawa metric is not bounded below. This is obtained by finding a direction along which the Sibony metric tends to infinity as the base point tends to the boundary. The analogous statement fails for a Lipschitz boundary. For a general C1 boundary, we give estimates for the Sibony metric in terms of some directional distance functions. For bounded pseudoconvex domains, the Blocki-Zwonek Suita-type theorem implies growth to infinity of the Bergman kernel; the fact that the Bergman kernel grows as the square of the reciprocal of the distance to the boundary, proved by S. Fu in the C2 case, is extended to bounded pseudoconvex domains with Lipschitz boundaries.

This survey shows how, for the Nevanlinna class N of the unit disc, one can define and often characterize the analogues of well-known objects and properties related to the algebra of bounded analytic functions $ H^\infty$: interpolating sequences, Corona theorem, sets of determination, stable rank, as well as the more recent notions of Weak Embedding Property and threshold of invertibility for quotient algebras. The general rule we observe is that a given result for $H^\infty$ can be transposed to N by replacing uniform bounds by a suitable control by positive harmonic functions. We show several instances where this rule applies, as well as some exceptions. We also briefly discuss the situation for the related Smirnov class.

We use a special version of the Corona Theorem in several variables, valid when all but one of the data functions are smooth, to generalize to the polydisc and to the ball results obtained by El Fallah, Kellay and Seip about cyclicity of non vanishing bounded holomorphic functions in large enough Banach spaces of analytic functions determined either by weighted sums of powers of Taylor coefficients or by radially weighted integrals of powers of the modulus of the function.

In the spirit of Kobayashi’s applications of methods of invariant metrics to questions of projective geometry, we introduce a projective analogue of the complex squeezing function. Using Frankel’s work, we prove that for convex domains it stays uniformly bounded from below. In the case of strongly convex domains, we show that it tends to 1 at the boundary. This is applied to get a new proof of a projective analogue of the Wong–Rosay theorem.

This survey shows how, for the Nevanlinna class 𝒩 of the unit disc, one can define and often characterize the analogues of well-known objects and properties related to the algebra of bounded analytic functions ℋ∞: interpolating sequences, Corona theorem, sets of determination, stable rank, as well as the more recent notions of Weak Embedding Property and threshold of invertibility for quotient algebras. The general rule we observe is that a given result for ℋ∞ can be transposed to 𝒩 by replacing uniform bounds by a suitable control by positive harmonic functions. We show several instances where this rule applies, as well as some exceptions. We also briefly discuss the situation for the related Smirnov class.