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Disks in the Euclidean plane

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May 4, 2021
linear algebra

Problem Statement

Let {D1,D2,...,Dn} \{D_1, D_2, ..., D_n \} be a set of disks in the Euclidean plane. Let ai,j=S(Di∩Dj) a_ {i, j} = S (D_i \cap D_j) be the area of Di∩Dj D_i \cap D_j . Prove that ∑i=1n∑j=1nai,jxixj≥0 \sum_ {i = 1} ^ n \sum_ {j = 1} ^ n a_ {i, j} x_ix_j \geq 0 for any real numbers x1,x2,...,xn x_1, x_2, ..., x_n .
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