An improved stability result for Grünbaum's inequality

Given a hyperplane $H$ cutting a compact, convex body $K$ of positive Lebesgue measure through its centroid, Grünbaum proved that $$\frac{\mathrm{\mid }K\cap H^{+}\mathrm{\mid }}{\mathrm{\mid }K\mathrm{\mid }}\ge {\left(\frac{n}{n+1}\right)}^n,$$ where $H^{+}$ is a half-space of boundary $H$. The inequality is sharp and equality is reached only if $K$ is a cone. Moreover, bodies that almost achieve equality are geometrically close to being cones, as Groemer showed in 2000 by giving his stability estimates for Grünbaum’s inequality. In this paper, we improve the exponent in the stability inequality from Groemer’s $\frac{1}{2n^2}$ to $\frac{1}{2n}$.