Grünbaum's theorem

For any convex set $\mathcal{S}$ with center-of-gravity $\mathbf{c}$, and any halfspace $\mathcal{Z}=\{\mathbf{x}|\langle\mathbf{a},\mathbf{x}-\mathbf{c}\rangle\leq 0\}$ then: $$\frac{\mathrm{vol}(\mathcal{S}\cap\mathcal{Z})}{\mathrm{vol}(\mathcal{S})} \geq \frac{1}{e} \approx .368$$

For any convex set $K \in \mathbb{R}^n$ with a center of gravity $c \in \mathbb{R}^n$, and any halfspace $H = \{x | a^\intercal (x − c) ≥ 0\}$ passing through $c$, $$\frac{1}{e} \leq \frac{\mathrm{vol}(K\cap H)}{\mathrm{vol}(K)} \leq \left(1-\frac{1}{e}\right)$$

see: Center-of-gravity method

References:

1. B. Grünbaum. Partitions of mass-distributions and of convex bodies by hyperplanes. Pacific J. Math., 10:1257–1261, 1960
2. https://www.cs.cmu.edu/~anupamg/advalgos17/scribes/lec16.pdf
3. https://user.math.uni-bremen.de/~grimpen/papers/grunbaum.pdf