Let $1
My answer : By using Minkowski inequality, we get $||tx+(1-t)y||_p\leq t||x||_p+(1-t)||y||_p=t+(1-t)=1$. But I don't get the strict inequality.
But,
For $p=2$: \begin{eqnarray}
||tx+(1-t)y||_2^2&=&t^2||x||_2^2+(1-t)^2||y||_2^2+2t(1-t)\Re(
Since $x\neq y$ and $||x||_2=||y||_2$, we conclude that $x\neq ky$ for every scalar hence we get $\Re(
Thanks everyone.