The marketing mix that relates to the product is price, product, place and promotion. A high end luxury public product is a product that people aspire to buy and typically increase social status.
Price refers to how the business will price it's products, with a high end public luxury product you would have a high price. Product refers to the quality and tangibility of the product (i.e. what it feels like, what it smells like, etc), this is where the business will differentiate itself from it's competition. With a high end public luxury product, the product would feel luxurious and use materials that others relate to success and luxury.
Place refers to the distribution of the product, how the consumer will get the product and with a high end public luxury product you would find it in shops such as Stone Island or an Apple store. Promotion refers to the promotional mix (advertising, sales promotion, PR, direct marketing and personal selling. With a high end public luxury product, advertising and personal selling would apply the most.