Punctured-Neighborhood Star-Convexity in Banach Spaces

• The paper demonstrates that metric balls in Banach spaces maintain star-convexity through quasihyperbolic and distance‐ratio metrics even after domain puncturing.
• It employs radial contraction, path interpolation, and analytic bounds to establish starlikeness in both full and punctured domain settings.
• The analysis reveals limitations in guaranteeing convexity, prompting further exploration of quantitative convexity measures in complex Banach spaces.

Punctured-neighborhood generalized star-convexity concerns the geometric properties of metric balls—specifically, quasihyperbolic and distance-ratio metric balls—on Banach spaces with or without punctures (i.e., domains excluding a point such as the origin). The core focus is on the starlikeness (star-convexity) of these balls: the condition that every straight segment from a distinguished center within the ball to any other point of the ball remains entirely inside the ball. Results in this area connect the structure of the metric, the geometry of the underlying Banach space, and the effect of puncturing the domain with respect to such generalized star-convexity properties (Rasila et al., 2010).

1. Quasihyperbolic and Distance-Ratio Metrics

Let $X$ denote a real Banach space and $\Omega \subsetneq X$ a domain. For each $x \in \Omega$, the boundary distance is $d(x) = \mathrm{dist}(x, \partial \Omega)$. Two central metrics are used:

• Quasihyperbolic metric: For a rectifiable path $\gamma\colon [0,1] \to \Omega$, its quasihyperbolic length is

$$\ell_k(\gamma) = \int_0^1 \frac{ \| d\gamma(t) \| }{ d(\gamma(t)) }.$$

The quasihyperbolic distance between $x, y \in \Omega$ is

$$k_\Omega(x, y) = \inf \{ \ell_k(\gamma) : \gamma \text{ joins } x \text{ to } y \text{ in } \Omega \}.$$

Associated balls are

$$B_k(x, r) = \{ y \in \Omega : k_\Omega(x, y) \le r \}.$$

• Distance-ratio (j) metric: Defined as

$$j_\Omega(x, y) = \log\left(1 + \frac{ \|x-y\| }{ d(x) \wedge d(y) } \right),$$

with balls

$$B_j(x, r) = \{ y \in \Omega : j_\Omega(x, y) \le r \}.$$

2. Definition and Characterization of Star-Convexity

A set $S \subset X$ is starlike with respect to $p \in S$ if

$$\forall y \in S,\quad [p, y] = \{ tp + (1-t)y : 0 \le t \le 1 \} \subset S.$$

For metric balls, this means that for each $y$ in the ball, the segment connecting the center to $y$ remains inside the ball. This property is pivotal when considering metric balls in both standard and punctured domains.

3. Generalized (“Punctured-Neighborhood”) Star-Convexity

“Generalized star-convexity” in the context of punctured neighborhoods does not introduce a new notion beyond the established starlikeness definition. In the punctured setting, e.g., $\Omega = X \setminus \{0\}$, the requirement remains that for all $y$ in the ball, every line segment from the center to $y$ lies entirely within the ball—even though the ambient domain omits a point. All results for generalized star-convexity are direct extensions of this standard criterion (Rasila et al., 2010).

4. Starlikeness Results for Metric Balls in Banach Spaces

Several precise theorems characterize when starlikeness holds for metric balls:

• Distance-ratio (j) balls: For any Banach space $X$ and proper subdomain $\Omega$, the ball $B_j(x_0, r)$ is starlike with respect to its center $x_0$ for all $r \le \log 2$. This follows by observing that if $j(x_0, y) \le \log 2$, then for any intermediate point along the segment $[x_0, y]$, the $j$-distance does not exceed $\log 2$.
• Balls in starlike domains: For starlike domains $\Omega$ with respect to $x_0$, both $B_k(x_0, r)$ and $B_j(x_0, r)$ are starlike centered at $x_0$ for all $r > 0$. In particular, the punctured space $\Omega = X \setminus \{0\}$ is starlike with respect to 0, so every such ball centered at 0 is starlike.
• Convexity results: When $\Omega$ is convex, for $x \in \Omega$ and $r > 0$, the balls $B_k(x, r)$ and $B_j(x, r)$ are convex. The proof leverages path-averaging arguments and convexity properties of the involved distances.

A critical remark concerns the lack of a universal positive "critical radius" for convexity: in the punctured case, unlike convexity, starlikeness persists from the center, but convexity for small balls cannot be guaranteed universally even in favorable Banach-space settings (Rasila et al., 2010).

5. Outline of Key Proof Techniques

• Starlikeness for j-balls: The main mechanism is to show that, given $j(x_0, y) \le \log 2$, it follows that the segment $[x_0, y]$ remains inside the ball. At each point, the defining ratio stays within its required bound, preserving starlikeness.
• Starlikeness for k-balls in starlike domains: Paths from $x_0$ to $y$ are "radially shrunk" towards $x_0$, and the quasihyperbolic length does not increase under this transformation due to the starlikeness of $\Omega$.
• Convexity (for convex $\Omega$): Combining two minimal-length paths with convex weights and using the convexity of the function $1/d(\cdot)$ shows that linear interpolation of endpoints yields points still within the ball.

6. Implications and Open Problems

The starlikeness of metric balls with respect to their centers is robust under puncturing for both $k$- and $j$-metrics. However, full convexity is more delicate in punctured spaces; no general convexity radius exists, notably in Banach spaces with strong geometric properties such as reflexivity, strict convexity, or smoothness. The existence of a quantitative convexity radius in such settings remains unresolved; the study ends with technical lemmas about near-convexity under additional assumptions, highlighting the boundary of current understanding (Rasila et al., 2010).