# Sum of squared nearest-neighbor distances between points on the sides of a rectangle

For positive real numbers $$a,b$$, let $$R$$ denote the $$a\times b$$ rectangle $$[0,a]\times[0,b]$$. Let $$A_1,\dots,A_4$$ be points on the sides of $$R$$, one point on each side. For each $$j=1,\dots,4$$, let $$s_j:=\min\{\|A_j-A_k\|^2\colon k\ne j\}$$.

Is it then always true that $$s_1+\dots+s_4\le2a^2+2b^2?$$

Here one may also assume that each of the four $$\frac a2\times \frac b2$$ rectangles $$R_1:=[0,\frac a2]\times[0,\frac b2],R_2:=[\frac a2,a]\times[0,\frac b2], R_3:=[0,\frac a2]\times[\frac b2,b],R_4:=[\frac a2,a]\times[\frac b2,b]$$ contains exactly one of the points $$A_1,\dots,A_4$$.

This question is what mainly remains to complete the answer to that related question.

• Soon after posting this question I understood how to get through the bottleneck to get an answer to this question and thus complete the answer to the mentioned related question at mathoverflow.net/questions/321033/… (which is what I have now done). – Iosif Pinelis Jan 21 '19 at 2:21