Chef's Delight

You are chef for a day. There are $1024$ different dishes you can make, but you need to pick $10$ of them for the dinner menu. You have sent all $3775$ attendees the list of dishes, and each attendee has ranked all $1024$ dishes in order of preference (from $1$ to $1024$, no ties).

You want to choose a list $L$ of $10$ dishes with the following property: for every dish $d$ that is not on $L$, there is some dish on $L$ that a majority of the attendees rank above $d$.

Can you always make such a list, no matter how the attendees rank the dishes?