Minimal size of share in Shamir's Sharing Scheme Why does the combined shared key in Adi Shamir's Sharing Scheme have to be smaller or equal than each secret share?
As listed by Shamir, the first property is:

(1) The size of each piece does not exceed the size of the original data. 


Here's what he means; there's a public value $p$; the original data is encoded as a value between 0 and $p-1$.  In addition, the secret portion of each share ('piece')  is a value between 0 and $p-1$, hence the size of each piece does not exceed the size of the original data.

What Shamir meant is that the scheme doesn't expand the share size beyond the size of the representation of the secret.  It can be shown that if the share size is smaller than the representation of the secret, then someone with $k-1$ shares (where $k$ is the threshold) can deduce some information about the secret (at the very least, some values which the secret cannot be).  Hence, Shamir's method is optimal in this sense.