Jishnu4414l wrote: The sum of squares of four reals $x,y,z,u$ is $1$. Find the minimum value of the expression $E=(x-y)(y-z)(z-u)(u-x)$. Find also the minimum values of $x$, $y$, $z$ and $u$ when this minimum occurs. Given that $a^2+b^2+c^2+d^2=1$, (1)find the minimum value of $(a-b)(b-c)(c-d)(d-a)$ when $a,b,c,d\geq 0$. (2)find the minimum value of $(a-b)(b-c)(c-d)(d-a)$. Let $a,b,c,d$ be real number such that $a^2 + b^2 + c^2 + d^2 = 1$. Prove that $$(a - b)(c - d) \geq -1$$

Let $a,b,c,d$ be real number such that $a^2 + b^2 + c^2 + d^2 = 1$. Prove that$$(a-b)(b-c)(c-d)(d-a) \geq -\frac{1}{8}$$Equality holds when $a=-b=\frac{1+\sqrt 3}{4},c=-d=\frac{1-\sqrt 3}{4} $ or ...