For a set A of points in the plane, not all collinear, we denote by $${\mathrm{tr}}(A)$$ tr ( A ) the number of triangles in a triangulation of A , that is, $${\mathrm{tr}}(A)=2i+b-2$$ tr ( A ) = 2 i + b - 2 , where b and i are the numbers of boundary and interior points of the convex hull [ A ] of A respectively. We conjecture the following discrete analog of the Brunn–Minkowski inequality: for any two finite point sets $$A,B\subset {{\mathbb {R}}}^2$$ A , B ⊂ R 2 one has $$\begin{aligned} {\mathrm{tr}}(A+B)\ge {\mathrm{tr}}(A)^{1/2}+{\mathrm{tr}}(B)^{1/2}. \end{aligned}$$ tr ( A + B ) ≥ tr ( A ) 1 / 2 + tr ( B ) 1 / 2 . We prove this conjecture in the cases where $$[A]=[B]$$ [ A ] = [ B ] , $$B=A\cup \{b\}$$ B = A ∪ { b } , $$|B|=3$$ | B | = 3 and if A and B have no interior points. A generalization to larger dimensions is also discussed.

中文翻译：

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平面上的三角剖分和离散布伦-闵可夫斯基不等式

对于平面中的一组 A 点，并非全部共线，我们用 $${\mathrm{tr}}(A)$$ tr ( A ) 表示 A 的三角剖分中的三角形数，即 $$ {\mathrm{tr}}(A)=2i+b-2$$ tr ( A ) = 2 i + b - 2 ，其中 b 和 i 是凸包 [ A ] 的边界点数和内点数A 分别。我们推测以下离散模拟的 Brunn-Minkowski 不等式：对于任何两个有限点集 $$A,B\subset {{\mathbb {R}}}^2$$ A , B ⊂ R 2 一个有 $$\开始{对齐} {\mathrm{tr}}(A+B)\ge {\mathrm{tr}}(A)^{1/2}+{\mathrm{tr}}(B)^{1/2 }. \end{对齐}$$ tr ( A + B ) ≥ tr ( A ) 1 / 2 + tr ( B ) 1 / 2 。我们在 $$[A]=[B]$$ [ A ] = [ B ] , $$B=A\cup \{b\}$$ B = A ∪ { b } , $ $|B|=3$$ | 乙 | = 3 并且如果 A 和 B 没有内点。还讨论了对更大维度的推广。