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We had a lot fun designing a potential solution. Simply check in an empty suitcase! But it would be very inconvenient carrying an empty suitcase just to check it in. So how about an inflatable suitcase! A self blow-up suitcase would be too inconvenient so it would have to contain a cartridge which you cracked open, mixing two chemicals and causing an expansion of gases. It could double as something to sit on too.
Airline companies have limits on the heaviness of luggage but we can't imagine they have limits on its lightness. But lightness could cause problems. We wouldn't want an unsuspecting luggage-handler falling over backwards trying to lift super-light air-luggage imagining it was normal heaviness. So we might have to put on stickers "Warning: Very Light". If the airlines introduced a minimum weight we might have to add weights pre-sealed inside the air-luggage. We could add words cast in metal to give the x-ray machine handlers something to read!
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Naturally the air-luggage would come pre-manufactured with a fake luggage tag. The address on all the tags would be the same - the factory where the air-luggage was made. That way we could reuse and recycle, reducing our costs and increasing our profits.
As a bonus we would actively look into ways of increasing the chance the airline lost your air-luggage. Each piece of air-luggage would come with lost-luggage claims-forms for all major airlines. These would be pre-filled-in by a sophisticated computer program enabling you to make a claim for all the lost air-items your air-luggage contained.