Photo by stayandroam from unsplash
Geometric and functional Brunn-Minkowski type inequalities for the lattice point enumerator $\mathrm{G}_n(\cdot)$ are provided. In particular, we show that $$\mathrm{G}_n((1-\lambda)K + \lambda L + (-1,1)^n)^{1/n}\geq (1-\lambda)\mathrm{G}_n(K)^{1/n}+\lambda\mathrm{G}_n(L)^{1/n}$$ for any non-empty bounded… Click to show full abstract
Geometric and functional Brunn-Minkowski type inequalities for the lattice point enumerator $\mathrm{G}_n(\cdot)$ are provided. In particular, we show that $$\mathrm{G}_n((1-\lambda)K + \lambda L + (-1,1)^n)^{1/n}\geq (1-\lambda)\mathrm{G}_n(K)^{1/n}+\lambda\mathrm{G}_n(L)^{1/n}$$ for any non-empty bounded sets $K, L\subset\mathbb{R}^n$ and all $\lambda\in(0,1)$. We also show that these new discrete versions imply the classical results, and discuss some links with other related inequalities.
Journal Title: Advances in Mathematics
Year Published: 2020
Link to full text (if available)
Share on Social Media: Sign Up to like & get
recommendations!