Reposted from 'Ask a FAQ', which is only designed for very brief and general queries. 

Numerous people have been asked/told to stop filming or taking photos here in Leicester at the new Highcross shopping area. It's private land apparently, even though it's open to the public and is not an enclosed space (it's outdoors). Is a store manager a legal representative of the landowners? As in, do they have any authority to request that filming be stopped?

They may well do. It's impossible to know without knowing the agreement they have with their landlord, the owner. It would be worth asking the landowner directly whether they have any rules about photography. 

What are the 'rules' about filming on private land when that land is a) open to the public, and b) outside?

It is allowed unless or until there is some communication of prohibition by the landowner, eg a sign or someone telling you it is prohibited. If you then continue you commit a trespass and can be ejected using minimum force. More likely police will be called. NB your camera cannot be seized by the landowner, nor do you have to delete them. They remain your copyright. See the FAQ for more info.

Many of the restaurants and bars have glass walls or large glass windows so surely there can be no "expectation of privacy" if you are carrying out general filming/photography outside, i.e. not photographing a specific individual. Would that be a valid position/argument?

It's not clear cut. Although you'd have to be very unlucky for privacy to be an issue, it is steadily expanding thanks to legal precedents. A court case a couple of years ago was successfully brought against a photographer who photographed in a restaurant for a local paper. He had the owner's consent, but the judge agreed with a gent in the background that he was entitled to privacy while eating his lunch.

A more common problem is that shops are very nervous about competitor stores photographing their displays. This is a particular issue with fashion shops, as designs are frequently stolen this way. Also expect problems if photographing banks or building societies. They always think you can only be planning a robbery, which of course would be very illegal. None of this restriction of innocent photography has a leg to stand on, but CCTV and paparazzi have made people think first of photography as surveiilance and spying or intrusion rather than art or creative expression. If you're going to do anything sensitive, telling them first and letting them know who you are should help. But some will wrongly insist you can't do it. There isn't a happy solution to this. Persisting, even though within your rights, can easily turn into an encounter with police who have a whole list of possible offences at their disposal for spoling your day. Antisocial behaviour is the current handy catch-all. Best plan : use a point and shoot, look like a tourist and speak a foreign language.