Let $C$ be a convex simple closed plane curve. For any $\theta \in \mathbb{R}$, let $w(\theta)$ be the length of the orthogonal projection of $C$ to a line in direction $(\cos\theta,\sin\theta)$. In other words, $w(\theta)$ is the distance between a pair of parallel tangent lines of $C$ pointing in the direction $(\cos(\theta+\pi/2),\sin(\theta+\pi/2))$. Denote by $\kappa_{max}$ the maximum curvature of $C$.
Why do we have that $w(\theta)\geq 2/\kappa_{max}$ for any $\theta \in \mathbb{R}$?
Using the average width formula (Cauchy-Crofton formula) $$L(C)=\frac12\int_0^{2\pi}w(\theta)d\theta\,,$$ I can prove that the maximal width $W:=\max_{\theta \in \mathbb{R}} w(\theta)$ satisfies the inequality. Also, from the same formula, it follows that $\min_{\theta \in \mathbb{R}} w(\theta)\leq 2/\kappa_{min}$, where $\kappa_{min}$ is the minimum curvature of $C$.