Simple bounds for the inradius and $\varepsilon$-inner neighbourhood of a convex body

Abstract

In this short note, we show that the inradius of a convex body is comparable to its volume divided by its surface area. We also give a simple formula, in terms of its volume and inradius, that is comparable to the volume of its intersection with the $\varepsilon$-neighbourhood of its boundary, and provide an application of this to self-projective attractors with convex holes.

Figures (3)

• Figure 1: Proof of Proposition \ref{['prop-heron']}: the decomposition of $\Omega$ into a) $\{\Omega_S\}$ and b) $\{\Delta_S\}$. The largest inscribed ball is depicted in faint yellow, and the incentre with a dot.
• Figure 2: Illustration of Proposition \ref{['prop:scale-copy']}. The scale copy of $\Omega$ is in green, and $L_\varepsilon(\Omega)$ in yellow, for a certain polygon $\Omega$ and value of $\varepsilon \in (0, \mathop{\mathrm{in}}\nolimits(\Omega))$. The incentre is marked with a dot.
• Figure 3: Sketch of Proposition \ref{['prop-curves']}: $\mathop{\mathrm{vol}}\nolimits(L_\varepsilon(\Omega))$ is constrained to lie in the yellow region.

Theorems & Definitions (15)

• Definition 1
• Proposition 2
• proof : Proof of lower bound
• proof : Proof of upper bound
• proof : Proof that constants are optimal
• Corollary 3: aspegren
• Proposition 4
• proof
• Proposition 5
• Proposition 6