Shape constraints in multivariate regression In many applications the objective is to find a meaningful structure in a relationship between dependent and independent variables through a regression kind of model expressed by Y = m(X) + ; where m is the so called regression function and a random error with zero mean. Nowadays many statistical problems are high dimensional since it is easier to collect data and modern computing power allows to consider massive amounts of information. Complex relations are evolved and evaluated more and more. The present thesis tries to answer some questions in this context by proposing new multivariate estimates utilizing shape constraints. Structured relationships in multivariate setups have not been considered in the literature that much probably due to inherent problems like data sparseness. Regression analysis is concerned with fitting a curve to a finite set of points in a space. An estimated regression function enables to predict evaluate and interpret a relationship between an explanatory variable and a response variable. In numerous examples order restricted inference is a reasonable approach to analyze the regression function ...